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

Percentage Accurate: 70.7% → 76.9%

Time: 17.4s

Alternatives: 7

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: 70.7% 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: 76.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

3. Taylor expanded in t 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 79.4

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

5. Applied rewrites 79.4%

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

6. Final simplification 79.4%

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

Alternative 2: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := \frac{\frac{a}{-3}}{b}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4200000000:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, z, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ a (* b 3.0))) (t_2 (/ (/ a -3.0) b)))
   (if (<= t_1 -1e-24)
     t_2
     (if (<= t_1 4200000000.0)
       (* (cos (fma (* -0.3333333333333333 t) z y)) (* (sqrt x) 2.0))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a / (b * 3.0);
	double t_2 = (a / -3.0) / b;
	double tmp;
	if (t_1 <= -1e-24) {
		tmp = t_2;
	} else if (t_1 <= 4200000000.0) {
		tmp = cos(fma((-0.3333333333333333 * t), z, y)) * (sqrt(x) * 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a / Float64(b * 3.0))
	t_2 = Float64(Float64(a / -3.0) / b)
	tmp = 0.0
	if (t_1 <= -1e-24)
		tmp = t_2;
	elseif (t_1 <= 4200000000.0)
		tmp = Float64(cos(fma(Float64(-0.3333333333333333 * t), z, y)) * Float64(sqrt(x) * 2.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / -3.0), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-24], t$95$2, If[LessEqual[t$95$1, 4200000000.0], N[(N[Cos[N[(N[(-0.3333333333333333 * t), $MachinePrecision] * z + y), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 a (*.f64 b #s(literal 3 binary64))) < -9.99999999999999924e-25 or 4.2e9 < (/.f64 a (*.f64 b #s(literal 3 binary64)))

   1. Initial program 90.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-frac N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 89.9

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

   5. Applied rewrites 89.9%

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

   7. Applied rewrites 90.0%

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

   9. Applied rewrites 90.0%

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

   11. Applied rewrites 90.1%

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

   if -9.99999999999999924e-25 < (/.f64 a (*.f64 b #s(literal 3 binary64))) < 4.2e9

   1. Initial program 59.6%

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

   3. Taylor expanded in b 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 55.8%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a}{b \cdot 3} \leq -1 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{a}{-3}}{b}\\ \mathbf{elif}\;\frac{a}{b \cdot 3} \leq 4200000000:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-0.3333333333333333 \cdot t, z, y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{-3}}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

3. Taylor expanded in t 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 79.4

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

5. Applied rewrites 79.4%

   \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y} - \frac{a}{b \cdot 3} \]
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-frac2 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. metadata-eval N/A

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

   9. metadata-eval N/A

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

   10. div-inv N/A

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

   11. un-div-inv N/A

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

   12. clear-num N/A

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

   13. lift-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 79.3

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

7. Applied rewrites 79.3%

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

8. Final simplification 79.3%

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

Alternative 4: 76.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

3. Taylor expanded in t 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. sub-neg N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. associate-*r/ N/A

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

   9. associate-*l/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

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

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

   13. lower-*.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 79.3

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

5. Applied rewrites 79.3%

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

6. Final simplification 79.3%

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

Alternative 5: 50.7% 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 74.2%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 47.5

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

5. Applied rewrites 47.5%

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

7. Applied rewrites 47.5%

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

9. Applied rewrites 47.5%

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

11. Applied rewrites 47.6%

    \[\leadsto \frac{\frac{a}{-3}}{b} \]
12. Add Preprocessing

Alternative 6: 50.7% accurate, 9.4× speedup

Math

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

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(a / Float64(-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[(a / N[(-3.0 * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.2%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 47.5

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

5. Applied rewrites 47.5%

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

7. Applied rewrites 47.5%

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

Alternative 7: 50.6% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 47.5

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

5. Applied rewrites 47.5%

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

Developer Target 1: 74.4% 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 2024270 
(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))))