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

Percentage Accurate: 70.7% → 77.1%

Time: 19.9s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 70.7% 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: 77.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. associate-*l* 70.0%

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

   2. fma-neg 70.0%

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

   3. *-commutative 70.0%

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

   4. sub-neg 70.0%

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

   5. *-commutative 70.0%

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

   6. distribute-neg-frac2 70.0%

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

   7. metadata-eval 70.0%

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

   8. distribute-frac-neg 70.0%

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

   9. neg-mul-1 70.0%

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

   10. *-commutative 70.0%

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

   11. times-frac 70.4%

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

   12. metadata-eval 70.4%

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

3. Simplified 70.4%

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

5. Taylor expanded in z around 0 76.8%

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

   1. associate-*r/ 77.1%

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

   2. clear-num 77.1%

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

7. Applied egg-rr 77.1%

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

   1. clear-num 77.0%

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

   2. associate-/r/ 77.1%

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

   3. associate-/r* 77.2%

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

   4. metadata-eval 77.2%

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

9. Applied egg-rr 77.2%

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

10. Final simplification 77.2%

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

Alternative 2: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. *-commutative 70.0%

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

   2. *-commutative 70.0%

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

   3. *-commutative 70.0%

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

   4. *-commutative 70.0%

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

   5. associate-/l* 70.1%

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

   6. *-commutative 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in z around 0 76.8%

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

6. Taylor expanded in a around 0 76.8%

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

   1. *-commutative 76.8%

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

   2. associate-*l/ 77.1%

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

   3. associate-/l* 77.1%

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

8. Simplified 77.1%

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

9. Final simplification 77.1%

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

Alternative 3: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, \sqrt{x}, \frac{1}{\frac{-3}{a} \cdot b}\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. associate-*l* 70.0%

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

   2. fma-neg 70.0%

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

   3. *-commutative 70.0%

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

   4. sub-neg 70.0%

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

   5. *-commutative 70.0%

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

   6. distribute-neg-frac2 70.0%

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

   7. metadata-eval 70.0%

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

   8. distribute-frac-neg 70.0%

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

   9. neg-mul-1 70.0%

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

   10. *-commutative 70.0%

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

   11. times-frac 70.4%

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

   12. metadata-eval 70.4%

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

3. Simplified 70.4%

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

5. Taylor expanded in z around 0 76.8%

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

   1. associate-*r/ 77.1%

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

   2. clear-num 77.1%

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

7. Applied egg-rr 77.1%

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

   1. clear-num 77.0%

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

   2. associate-/r/ 77.1%

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

   3. associate-/r* 77.2%

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

   4. metadata-eval 77.2%

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

9. Applied egg-rr 77.2%

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

10. Taylor expanded in y around 0 69.0%

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

11. Final simplification 69.0%

    \[\leadsto \mathsf{fma}\left(2, \sqrt{x}, \frac{1}{\frac{-3}{a} \cdot b}\right) \]
12. Add Preprocessing

Alternative 4: 65.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (2.0d0 * sqrt(x)) - (0.3333333333333333d0 * (a / b))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return (2.0 * Math.sqrt(x)) - (0.3333333333333333 * (a / b));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. *-commutative 70.0%

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

   2. *-commutative 70.0%

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

   3. *-commutative 70.0%

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

   4. *-commutative 70.0%

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

   5. associate-/l* 70.1%

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

   6. *-commutative 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in z around 0 76.8%

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

6. Taylor expanded in y around 0 68.7%

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

7. Final simplification 68.7%

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

Alternative 5: 50.7% accurate, 43.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. associate-*l* 70.0%

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

   2. fma-neg 70.0%

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

   3. *-commutative 70.0%

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

   4. sub-neg 70.0%

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

   5. *-commutative 70.0%

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

   6. distribute-neg-frac2 70.0%

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

   7. metadata-eval 70.0%

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

   8. distribute-frac-neg 70.0%

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

   9. neg-mul-1 70.0%

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

   10. *-commutative 70.0%

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

   11. times-frac 70.4%

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

   12. metadata-eval 70.4%

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

3. Simplified 70.4%

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

5. Taylor expanded in z around 0 76.8%

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

6. Taylor expanded in a around inf 53.5%

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

7. Final simplification 53.5%

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

Alternative 6: 50.8% accurate, 43.4× 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 70.0%

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

   1. associate-*l* 70.0%

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

   2. fma-neg 70.0%

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

   3. *-commutative 70.0%

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

   4. sub-neg 70.0%

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

   5. *-commutative 70.0%

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

   6. distribute-neg-frac2 70.0%

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

   7. metadata-eval 70.0%

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

   8. distribute-frac-neg 70.0%

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

   9. neg-mul-1 70.0%

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

   10. *-commutative 70.0%

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

   11. times-frac 70.4%

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

   12. metadata-eval 70.4%

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

3. Simplified 70.4%

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

5. Taylor expanded in z around 0 76.8%

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

6. Taylor expanded in a around inf 53.5%

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

   1. clear-num 53.4%

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

   2. un-div-inv 53.7%

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

8. Applied egg-rr 53.7%

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

   1. associate-/r/ 53.7%

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

10. Simplified 53.7%

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

11. Final simplification 53.7%

    \[\leadsto a \cdot \frac{-0.3333333333333333}{b} \]
12. Add Preprocessing

Alternative 7: 50.8% accurate, 43.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. associate-*l* 70.0%

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

   2. fma-neg 70.0%

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

   3. *-commutative 70.0%

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

   4. sub-neg 70.0%

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

   5. *-commutative 70.0%

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

   6. distribute-neg-frac2 70.0%

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

   7. metadata-eval 70.0%

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

   8. distribute-frac-neg 70.0%

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

   9. neg-mul-1 70.0%

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

   10. *-commutative 70.0%

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

   11. times-frac 70.4%

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

   12. metadata-eval 70.4%

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

3. Simplified 70.4%

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

5. Taylor expanded in z around 0 76.8%

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

6. Taylor expanded in a around inf 53.5%

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

   1. clear-num 53.4%

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

   2. un-div-inv 53.7%

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

8. Applied egg-rr 53.7%

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

9. Final simplification 53.7%

   \[\leadsto \frac{-0.3333333333333333}{\frac{b}{a}} \]
10. Add Preprocessing

Alternative 8: 50.8% accurate, 43.4× speedup

Math

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

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(Float64(a * -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[(N[(a * -0.3333333333333333), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 70.0%

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

   1. associate-*l* 70.0%

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

   2. fma-neg 70.0%

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

   3. *-commutative 70.0%

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

   4. sub-neg 70.0%

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

   5. *-commutative 70.0%

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

   6. distribute-neg-frac2 70.0%

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

   7. metadata-eval 70.0%

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

   8. distribute-frac-neg 70.0%

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

   9. neg-mul-1 70.0%

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

   10. *-commutative 70.0%

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

   11. times-frac 70.4%

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

   12. metadata-eval 70.4%

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

3. Simplified 70.4%

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

5. Taylor expanded in z around 0 76.8%

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

6. Taylor expanded in a around inf 53.5%

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

   1. clear-num 53.4%

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

   2. un-div-inv 53.7%

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

8. Applied egg-rr 53.7%

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

   1. associate-/r/ 53.7%

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

   2. associate-*l/ 53.7%

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

10. Simplified 53.7%

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

11. Final simplification 53.7%

    \[\leadsto \frac{a \cdot -0.3333333333333333}{b} \]
12. Add Preprocessing

Alternative 9: 50.9% accurate, 43.4× 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 70.0%

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

   1. associate-*l* 70.0%

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

   2. fma-neg 70.0%

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

   3. *-commutative 70.0%

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

   4. sub-neg 70.0%

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

   5. *-commutative 70.0%

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

   6. distribute-neg-frac2 70.0%

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

   7. metadata-eval 70.0%

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

   8. distribute-frac-neg 70.0%

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

   9. neg-mul-1 70.0%

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

   10. *-commutative 70.0%

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

   11. times-frac 70.4%

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

   12. metadata-eval 70.4%

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

3. Simplified 70.4%

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

5. Taylor expanded in z around 0 76.8%

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

6. Taylor expanded in a around inf 53.5%

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

   1. metadata-eval 53.5%

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

   2. times-frac 53.4%

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

   3. *-lft-identity 53.4%

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

   4. associate-/r* 53.9%

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

8. Simplified 53.9%

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

9. Final simplification 53.9%

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

Developer target: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{0.3333333333333333}{z}}{t}\\ t_2 := \frac{\frac{a}{3}}{b}\\ t_3 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;t\_3 \cdot \cos \left(\frac{1}{y} - t\_1\right) - t\_2\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - t\_2\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - t\_1\right) \cdot t\_3 - \frac{\frac{a}{b}}{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (/ 0.3333333333333333 z) t))
        (t_2 (/ (/ a 3.0) b))
        (t_3 (* 2.0 (sqrt x))))
   (if (< z -1.3793337487235141e+129)
     (- (* t_3 (cos (- (/ 1.0 y) t_1))) t_2)
     (if (< z 3.516290613555987e+106)
       (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) t_2)
       (- (* (cos (- y t_1)) t_3) (/ (/ a b) 3.0))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (0.3333333333333333d0 / z) / t
    t_2 = (a / 3.0d0) / b
    t_3 = 2.0d0 * sqrt(x)
    if (z < (-1.3793337487235141d+129)) then
        tmp = (t_3 * cos(((1.0d0 / y) - t_1))) - t_2
    else if (z < 3.516290613555987d+106) then
        tmp = ((sqrt(x) * 2.0d0) * cos((y - ((t / 3.0d0) * z)))) - t_2
    else
        tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (0.3333333333333333 / z) / t;
	double t_2 = (a / 3.0) / b;
	double t_3 = 2.0 * Math.sqrt(x);
	double tmp;
	if (z < -1.3793337487235141e+129) {
		tmp = (t_3 * Math.cos(((1.0 / y) - t_1))) - t_2;
	} else if (z < 3.516290613555987e+106) {
		tmp = ((Math.sqrt(x) * 2.0) * Math.cos((y - ((t / 3.0) * z)))) - t_2;
	} else {
		tmp = (Math.cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (0.3333333333333333 / z) / t
	t_2 = (a / 3.0) / b
	t_3 = 2.0 * math.sqrt(x)
	tmp = 0
	if z < -1.3793337487235141e+129:
		tmp = (t_3 * math.cos(((1.0 / y) - t_1))) - t_2
	elif z < 3.516290613555987e+106:
		tmp = ((math.sqrt(x) * 2.0) * math.cos((y - ((t / 3.0) * z)))) - t_2
	else:
		tmp = (math.cos((y - t_1)) * t_3) - ((a / b) / 3.0)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(0.3333333333333333 / z) / t)
	t_2 = Float64(Float64(a / 3.0) / b)
	t_3 = Float64(2.0 * sqrt(x))
	tmp = 0.0
	if (z < -1.3793337487235141e+129)
		tmp = Float64(Float64(t_3 * cos(Float64(Float64(1.0 / y) - t_1))) - t_2);
	elseif (z < 3.516290613555987e+106)
		tmp = Float64(Float64(Float64(sqrt(x) * 2.0) * cos(Float64(y - Float64(Float64(t / 3.0) * z)))) - t_2);
	else
		tmp = Float64(Float64(cos(Float64(y - t_1)) * t_3) - Float64(Float64(a / b) / 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (0.3333333333333333 / z) / t;
	t_2 = (a / 3.0) / b;
	t_3 = 2.0 * sqrt(x);
	tmp = 0.0;
	if (z < -1.3793337487235141e+129)
		tmp = (t_3 * cos(((1.0 / y) - t_1))) - t_2;
	elseif (z < 3.516290613555987e+106)
		tmp = ((sqrt(x) * 2.0) * cos((y - ((t / 3.0) * z)))) - t_2;
	else
		tmp = (cos((y - t_1)) * t_3) - ((a / b) / 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(0.3333333333333333 / z), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / 3.0), $MachinePrecision] / b), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.3793337487235141e+129], N[(N[(t$95$3 * N[Cos[N[(N[(1.0 / y), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], If[Less[z, 3.516290613555987e+106], N[(N[(N[(N[Sqrt[x], $MachinePrecision] * 2.0), $MachinePrecision] * N[Cos[N[(y - N[(N[(t / 3.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[(N[Cos[N[(y - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(N[(a / b), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024053 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :alt
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

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