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

Percentage Accurate: 69.4% → 75.9%

Time: 18.8s

Alternatives: 9

Speedup: 1.2×

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: 69.4% 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: 75.9% accurate, 1.2× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in z around 0

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

   1. lower-cos.f64 75.3

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

5. Applied rewrites 75.3%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. distribute-neg-frac2 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 75.3

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

7. Applied rewrites 75.3%

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

Alternative 2: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := 2 \cdot \sqrt{x}\\ t_3 := t\_2 - t\_1\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-128}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-97}:\\ \;\;\;\;\cos y \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (* 2.0 (sqrt x))) (t_3 (- t_2 t_1)))
   (if (<= t_1 -2e-128) t_3 (if (<= t_1 1e-97) (* (cos y) t_2) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = 2.0 * sqrt(x);
	double t_3 = t_2 - t_1;
	double tmp;
	if (t_1 <= -2e-128) {
		tmp = t_3;
	} else if (t_1 <= 1e-97) {
		tmp = cos(y) * t_2;
	} else {
		tmp = t_3;
	}
	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 = a / (3.0d0 * b)
    t_2 = 2.0d0 * sqrt(x)
    t_3 = t_2 - t_1
    if (t_1 <= (-2d-128)) then
        tmp = t_3
    else if (t_1 <= 1d-97) then
        tmp = cos(y) * t_2
    else
        tmp = t_3
    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 t_2 = 2.0 * Math.sqrt(x);
	double t_3 = t_2 - t_1;
	double tmp;
	if (t_1 <= -2e-128) {
		tmp = t_3;
	} else if (t_1 <= 1e-97) {
		tmp = Math.cos(y) * t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = 2.0 * math.sqrt(x)
	t_3 = t_2 - t_1
	tmp = 0
	if t_1 <= -2e-128:
		tmp = t_3
	elif t_1 <= 1e-97:
		tmp = math.cos(y) * t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(2.0 * sqrt(x))
	t_3 = Float64(t_2 - t_1)
	tmp = 0.0
	if (t_1 <= -2e-128)
		tmp = t_3;
	elseif (t_1 <= 1e-97)
		tmp = Float64(cos(y) * t_2);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	t_2 = 2.0 * sqrt(x);
	t_3 = t_2 - t_1;
	tmp = 0.0;
	if (t_1 <= -2e-128)
		tmp = t_3;
	elseif (t_1 <= 1e-97)
		tmp = cos(y) * t_2;
	else
		tmp = t_3;
	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]}, Block[{t$95$2 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-128], t$95$3, If[LessEqual[t$95$1, 1e-97], N[(N[Cos[y], $MachinePrecision] * t$95$2), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -2.00000000000000011e-128 or 1.00000000000000004e-97 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 72.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-cos.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 70.0

          \[\leadsto \mathsf{fma}\left(\sqrt{x}, \left(0.6666666666666666 \cdot \left(z \cdot \sin y\right)\right) \cdot t, \sqrt{x} \cdot \left(2 \cdot \color{blue}{\cos y}\right)\right) - \frac{a}{b \cdot 3} \]

   8. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \left(0.6666666666666666 \cdot \left(z \cdot \sin y\right)\right) \cdot t, \sqrt{x} \cdot \left(2 \cdot \cos y\right)\right)} - \frac{a}{b \cdot 3} \]

   9. Taylor expanded in y around 0

      \[\leadsto 2 \cdot \color{blue}{\sqrt{x}} - \frac{a}{b \cdot 3} \]
   10. Step-by-step derivation

   11. Applied rewrites 81.0%

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

   if -2.00000000000000011e-128 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 1.00000000000000004e-97

   1. Initial program 49.1%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. cos-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 48.5%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(z, t \cdot 0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(z \cdot t, -0.3333333333333333, y\right)\right), 0.5, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{2} \cdot \left(\cos \left(y + \frac{-1}{3} \cdot \left(t \cdot z\right)\right) + \cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right), \sin y \cdot \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 49.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -2 \cdot 10^{-128}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 10^{-97}:\\ \;\;\;\;\cos y \cdot \left(2 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in z around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 75.2

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

5. Applied rewrites 75.2%

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

6. Final simplification 75.2%

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

Alternative 4: 60.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{a}{-3}}{b}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (<= t_1 -1.5e-61)
     (/ (/ a -3.0) b)
     (if (<= t_1 2e-101) (* 2.0 (sqrt x)) (/ a (* b -3.0))))))

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 <= -1.5e-61) {
		tmp = (a / -3.0) / b;
	} else if (t_1 <= 2e-101) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = 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) :: tmp
    t_1 = a / (3.0d0 * b)
    if (t_1 <= (-1.5d-61)) then
        tmp = (a / (-3.0d0)) / b
    else if (t_1 <= 2d-101) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = 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 = a / (3.0 * b);
	double tmp;
	if (t_1 <= -1.5e-61) {
		tmp = (a / -3.0) / b;
	} else if (t_1 <= 2e-101) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = a / (b * -3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if t_1 <= -1.5e-61:
		tmp = (a / -3.0) / b
	elif t_1 <= 2e-101:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = a / (b * -3.0)
	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 <= -1.5e-61)
		tmp = Float64(Float64(a / -3.0) / b);
	elseif (t_1 <= 2e-101)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = Float64(a / Float64(b * -3.0));
	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 <= -1.5e-61)
		tmp = (a / -3.0) / b;
	elseif (t_1 <= 2e-101)
		tmp = 2.0 * sqrt(x);
	else
		tmp = a / (b * -3.0);
	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[LessEqual[t$95$1, -1.5e-61], N[(N[(a / -3.0), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[t$95$1, 2e-101], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.50000000000000006e-61

   1. Initial program 79.5%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 80.6%

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

   9. Applied rewrites 80.9%

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

   if -1.50000000000000006e-61 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.0000000000000001e-101

   1. Initial program 47.8%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. cos-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 47.7%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(z, t \cdot 0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(z \cdot t, -0.3333333333333333, y\right)\right), 0.5, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{2} \cdot \left(\cos \left(y + \frac{-1}{3} \cdot \left(t \cdot z\right)\right) + \cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right), \sin y \cdot \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 26.4%

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(y, \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right), \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

   11. Taylor expanded in t around 0

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 29.0%

       \[\leadsto \sqrt{x} \cdot 2 \]

   if 2.0000000000000001e-101 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 70.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\frac{a}{b \cdot -3}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{a}{-3}}{b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \frac{a}{b \cdot -3}\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (/ a (* b -3.0))))
   (if (<= t_1 -1.5e-61) t_2 (if (<= t_1 2e-101) (* 2.0 (sqrt x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = a / (b * -3.0);
	double tmp;
	if (t_1 <= -1.5e-61) {
		tmp = t_2;
	} else if (t_1 <= 2e-101) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = t_2;
	}
	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 = a / (3.0d0 * b)
    t_2 = a / (b * (-3.0d0))
    if (t_1 <= (-1.5d-61)) then
        tmp = t_2
    else if (t_1 <= 2d-101) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = t_2
    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 t_2 = a / (b * -3.0);
	double tmp;
	if (t_1 <= -1.5e-61) {
		tmp = t_2;
	} else if (t_1 <= 2e-101) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = a / (b * -3.0)
	tmp = 0
	if t_1 <= -1.5e-61:
		tmp = t_2
	elif t_1 <= 2e-101:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(a / Float64(b * -3.0))
	tmp = 0.0
	if (t_1 <= -1.5e-61)
		tmp = t_2;
	elseif (t_1 <= 2e-101)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	t_2 = a / (b * -3.0);
	tmp = 0.0;
	if (t_1 <= -1.5e-61)
		tmp = t_2;
	elseif (t_1 <= 2e-101)
		tmp = 2.0 * sqrt(x);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(a / N[(b * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.5e-61], t$95$2, If[LessEqual[t$95$1, 2e-101], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.50000000000000006e-61 or 2.0000000000000001e-101 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 75.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 78.2%

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

   if -1.50000000000000006e-61 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.0000000000000001e-101

   1. Initial program 47.8%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. cos-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 47.7%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(z, t \cdot 0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(z \cdot t, -0.3333333333333333, y\right)\right), 0.5, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{2} \cdot \left(\cos \left(y + \frac{-1}{3} \cdot \left(t \cdot z\right)\right) + \cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right), \sin y \cdot \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 26.4%

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(y, \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right), \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

   11. Taylor expanded in t around 0

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 29.0%

       \[\leadsto \sqrt{x} \cdot 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{b \cdot -3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{b} \cdot -0.3333333333333333\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))))
   (if (<= t_1 -1.5e-61)
     (* (/ a b) -0.3333333333333333)
     (if (<= t_1 2e-101) (* 2.0 (sqrt x)) (* a (/ -0.3333333333333333 b))))))

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 <= -1.5e-61) {
		tmp = (a / b) * -0.3333333333333333;
	} else if (t_1 <= 2e-101) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = a * (-0.3333333333333333 / b);
	}
	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 <= (-1.5d-61)) then
        tmp = (a / b) * (-0.3333333333333333d0)
    else if (t_1 <= 2d-101) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = a * ((-0.3333333333333333d0) / b)
    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 <= -1.5e-61) {
		tmp = (a / b) * -0.3333333333333333;
	} else if (t_1 <= 2e-101) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = a * (-0.3333333333333333 / b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	tmp = 0
	if t_1 <= -1.5e-61:
		tmp = (a / b) * -0.3333333333333333
	elif t_1 <= 2e-101:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = a * (-0.3333333333333333 / b)
	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 <= -1.5e-61)
		tmp = Float64(Float64(a / b) * -0.3333333333333333);
	elseif (t_1 <= 2e-101)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = Float64(a * Float64(-0.3333333333333333 / b));
	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 <= -1.5e-61)
		tmp = (a / b) * -0.3333333333333333;
	elseif (t_1 <= 2e-101)
		tmp = 2.0 * sqrt(x);
	else
		tmp = a * (-0.3333333333333333 / b);
	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[LessEqual[t$95$1, -1.5e-61], N[(N[(a / b), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], If[LessEqual[t$95$1, 2e-101], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(a * N[(-0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.50000000000000006e-61

   1. Initial program 79.5%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -1.50000000000000006e-61 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.0000000000000001e-101

   1. Initial program 47.8%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. cos-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 47.7%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(z, t \cdot 0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(z \cdot t, -0.3333333333333333, y\right)\right), 0.5, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{2} \cdot \left(\cos \left(y + \frac{-1}{3} \cdot \left(t \cdot z\right)\right) + \cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right), \sin y \cdot \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 26.4%

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(y, \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right), \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

   11. Taylor expanded in t around 0

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 29.0%

       \[\leadsto \sqrt{x} \cdot 2 \]

   if 2.0000000000000001e-101 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 70.6%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 75.4%

      \[\leadsto a \cdot \color{blue}{\frac{-0.3333333333333333}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{b} \cdot -0.3333333333333333\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b))) (t_2 (* a (/ -0.3333333333333333 b))))
   (if (<= t_1 -1.5e-61) t_2 (if (<= t_1 2e-101) (* 2.0 (sqrt x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (3.0 * b);
	double t_2 = a * (-0.3333333333333333 / b);
	double tmp;
	if (t_1 <= -1.5e-61) {
		tmp = t_2;
	} else if (t_1 <= 2e-101) {
		tmp = 2.0 * sqrt(x);
	} else {
		tmp = t_2;
	}
	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 = a / (3.0d0 * b)
    t_2 = a * ((-0.3333333333333333d0) / b)
    if (t_1 <= (-1.5d-61)) then
        tmp = t_2
    else if (t_1 <= 2d-101) then
        tmp = 2.0d0 * sqrt(x)
    else
        tmp = t_2
    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 t_2 = a * (-0.3333333333333333 / b);
	double tmp;
	if (t_1 <= -1.5e-61) {
		tmp = t_2;
	} else if (t_1 <= 2e-101) {
		tmp = 2.0 * Math.sqrt(x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a / (3.0 * b)
	t_2 = a * (-0.3333333333333333 / b)
	tmp = 0
	if t_1 <= -1.5e-61:
		tmp = t_2
	elif t_1 <= 2e-101:
		tmp = 2.0 * math.sqrt(x)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(3.0 * b))
	t_2 = Float64(a * Float64(-0.3333333333333333 / b))
	tmp = 0.0
	if (t_1 <= -1.5e-61)
		tmp = t_2;
	elseif (t_1 <= 2e-101)
		tmp = Float64(2.0 * sqrt(x));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a / (3.0 * b);
	t_2 = a * (-0.3333333333333333 / b);
	tmp = 0.0;
	if (t_1 <= -1.5e-61)
		tmp = t_2;
	elseif (t_1 <= 2e-101)
		tmp = 2.0 * sqrt(x);
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(a * N[(-0.3333333333333333 / b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.5e-61], t$95$2, If[LessEqual[t$95$1, 2e-101], N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -1.50000000000000006e-61 or 2.0000000000000001e-101 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 75.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{a}{b}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 78.0%

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

   if -1.50000000000000006e-61 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 2.0000000000000001e-101

   1. Initial program 47.8%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. cos-mult N/A

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

      5. div-inv N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 47.7%

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(z, t \cdot 0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(z \cdot t, -0.3333333333333333, y\right)\right), 0.5, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right)} - \frac{a}{b \cdot 3} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{2} \cdot \left(\cos \left(y + \frac{-1}{3} \cdot \left(t \cdot z\right)\right) + \cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 45.9%

      \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right), \sin y \cdot \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 26.4%

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(y, \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right), \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

   11. Taylor expanded in t around 0

       \[\leadsto 2 \cdot \sqrt{x} \]
   12. Step-by-step derivation

   13. Applied rewrites 29.0%

       \[\leadsto \sqrt{x} \cdot 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1.5 \cdot 10^{-61}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{-0.3333333333333333}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in z around 0

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

   1. lower-cos.f64 75.3

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

5. Applied rewrites 75.3%

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

6. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. lower-cos.f64 63.5

       \[\leadsto \mathsf{fma}\left(\sqrt{x}, \left(0.6666666666666666 \cdot \left(z \cdot \sin y\right)\right) \cdot t, \sqrt{x} \cdot \left(2 \cdot \color{blue}{\cos y}\right)\right) - \frac{a}{b \cdot 3} \]

8. Applied rewrites 63.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \left(0.6666666666666666 \cdot \left(z \cdot \sin y\right)\right) \cdot t, \sqrt{x} \cdot \left(2 \cdot \cos y\right)\right)} - \frac{a}{b \cdot 3} \]

9. Taylor expanded in y around 0

   \[\leadsto 2 \cdot \color{blue}{\sqrt{x}} - \frac{a}{b \cdot 3} \]
10. Step-by-step derivation

11. Applied rewrites 66.5%

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

12. Final simplification 66.5%

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

Alternative 9: 17.9% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \sqrt{x} \end{array} \]

FPCore

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

C

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

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)
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);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

   1. lift-cos.f64 N/A

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

   2. lift--.f64 N/A

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

   3. cos-diff N/A

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

   4. cos-mult N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 66.0%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(\mathsf{fma}\left(z, t \cdot 0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(z \cdot t, -0.3333333333333333, y\right)\right), 0.5, \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.3333333333333333\right)\right)\right)} - \frac{a}{b \cdot 3} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2 \cdot \left(\sqrt{x} \cdot \left(\frac{1}{2} \cdot \left(\cos \left(y + \frac{-1}{3} \cdot \left(t \cdot z\right)\right) + \cos \left(y + \frac{1}{3} \cdot \left(t \cdot z\right)\right)\right) + \sin y \cdot \sin \left(\frac{1}{3} \cdot \left(t \cdot z\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-fma.f64 N/A

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

7. Applied rewrites 22.9%

   \[\leadsto \color{blue}{\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right) + \cos \left(\mathsf{fma}\left(0.3333333333333333, t \cdot z, y\right)\right), \sin y \cdot \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

8. Taylor expanded in y around 0

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

10. Applied rewrites 14.0%

    \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(y, \sin \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right), \cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right)\right)} \]

11. Taylor expanded in t around 0

    \[\leadsto 2 \cdot \sqrt{x} \]
12. Step-by-step derivation

13. Applied rewrites 14.8%

    \[\leadsto \sqrt{x} \cdot 2 \]

14. Final simplification 14.8%

    \[\leadsto 2 \cdot \sqrt{x} \]
15. Add Preprocessing

Developer Target 1: 73.6% accurate, 0.8× 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 2024226 
(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))))