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

Percentage Accurate: 69.8% → 76.0%

Time: 20.6s

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: 69.8% 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: 76.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Taylor expanded in 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 74.1

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

5. Applied rewrites 74.1%

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

6. Final simplification 74.1%

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

Alternative 2: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-126}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b)))
        (t_2 (fma (/ a b) -0.3333333333333333 (* 2.0 (* (sqrt x) 1.0)))))
   (if (<= t_1 -1e-126)
     t_2
     (if (<= t_1 2e-94)
       (* (* 2.0 (sqrt x)) (cos (fma t (* z -0.3333333333333333) y)))
       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 = fma((a / b), -0.3333333333333333, (2.0 * (sqrt(x) * 1.0)));
	double tmp;
	if (t_1 <= -1e-126) {
		tmp = t_2;
	} else if (t_1 <= 2e-94) {
		tmp = (2.0 * sqrt(x)) * cos(fma(t, (z * -0.3333333333333333), y));
	} 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 = fma(Float64(a / b), -0.3333333333333333, Float64(2.0 * Float64(sqrt(x) * 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-126)
		tmp = t_2;
	elseif (t_1 <= 2e-94)
		tmp = Float64(Float64(2.0 * sqrt(x)) * cos(fma(t, Float64(z * -0.3333333333333333), y)));
	else
		tmp = t_2;
	end
	return 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[(N[(a / b), $MachinePrecision] * -0.3333333333333333 + N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-126], t$95$2, If[LessEqual[t$95$1, 2e-94], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(t * N[(z * -0.3333333333333333), $MachinePrecision] + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -9.9999999999999995e-127 or 1.9999999999999999e-94 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 75.2%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 83.7

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{a}{b}, \frac{-1}{3}, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.0%

       \[\leadsto \mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right) \]

   if -9.9999999999999995e-127 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 1.9999999999999999e-94

   1. Initial program 53.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 54.3%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right)\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\mathsf{fma}\left(t, z \cdot -0.3333333333333333, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{3 \cdot b}\\ t_2 := \mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-126}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* 3.0 b)))
        (t_2 (fma (/ a b) -0.3333333333333333 (* 2.0 (* (sqrt x) 1.0)))))
   (if (<= t_1 -1e-126)
     t_2
     (if (<= t_1 2e-94) (* (* 2.0 (sqrt x)) (cos y)) 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 = fma((a / b), -0.3333333333333333, (2.0 * (sqrt(x) * 1.0)));
	double tmp;
	if (t_1 <= -1e-126) {
		tmp = t_2;
	} else if (t_1 <= 2e-94) {
		tmp = (2.0 * sqrt(x)) * cos(y);
	} 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 = fma(Float64(a / b), -0.3333333333333333, Float64(2.0 * Float64(sqrt(x) * 1.0)))
	tmp = 0.0
	if (t_1 <= -1e-126)
		tmp = t_2;
	elseif (t_1 <= 2e-94)
		tmp = Float64(Float64(2.0 * sqrt(x)) * cos(y));
	else
		tmp = t_2;
	end
	return 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[(N[(a / b), $MachinePrecision] * -0.3333333333333333 + N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-126], t$95$2, If[LessEqual[t$95$1, 2e-94], N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -9.9999999999999995e-127 or 1.9999999999999999e-94 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 75.2%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. neg-mul-1 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. lift-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 83.7

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

   7. Applied rewrites 83.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{a}{b}, \frac{-1}{3}, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.0%

       \[\leadsto \mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right) \]

   if -9.9999999999999995e-127 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 1.9999999999999999e-94

   1. Initial program 53.1%

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
   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. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-cos.f64 N/A

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

      18. lower-cos.f64 54.6

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

      19. lift-/.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f64 N/A

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

      23. div-inv N/A

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

      24. lower-*.f64 N/A

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

      25. metadata-eval 55.1

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

   4. Applied rewrites 55.1%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. cos-neg N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-neg-in N/A

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

      12. cos-neg N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

   7. Applied rewrites 55.2%

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

   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 54.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{3 \cdot b} \leq -1 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right)\\ \mathbf{elif}\;\frac{a}{3 \cdot b} \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

3. Taylor expanded in 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 74.0

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

5. Applied rewrites 74.0%

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

Alternative 5: 65.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (fma (/ a b) -0.3333333333333333 (* 2.0 (* (sqrt x) 1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma((a / b), -0.3333333333333333, (2.0 * (sqrt(x) * 1.0)));
}

Julia

function code(x, y, z, t, a, b)
	return fma(Float64(a / b), -0.3333333333333333, Float64(2.0 * Float64(sqrt(x) * 1.0)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(a / b), $MachinePrecision] * -0.3333333333333333 + N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.0%

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

3. Taylor expanded in 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 74.1

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

5. Applied rewrites 74.1%

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

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac N/A

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

   6. neg-mul-1 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. metadata-eval N/A

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

   11. lift-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 74.0

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

7. Applied rewrites 74.0%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(\frac{a}{b}, \frac{-1}{3}, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 62.3%

    \[\leadsto \mathsf{fma}\left(\frac{a}{b}, -0.3333333333333333, 2 \cdot \left(\sqrt{x} \cdot 1\right)\right) \]
11. Add Preprocessing

Alternative 6: 50.6% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
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 46.8

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

5. Applied rewrites 46.8%

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

7. Applied rewrites 46.9%

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

9. Applied rewrites 46.9%

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

Alternative 7: 50.6% accurate, 9.4× 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 68.0%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
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 46.8

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

5. Applied rewrites 46.8%

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

7. Applied rewrites 46.9%

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

Alternative 8: 50.5% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
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 46.8

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

5. Applied rewrites 46.8%

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

Alternative 9: 50.6% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

   \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
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 46.8

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

5. Applied rewrites 46.8%

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

7. Applied rewrites 46.8%

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

Developer Target 1: 73.7% 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 2024232 
(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))))