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

Percentage Accurate: 99.8% → 99.8%

Time: 6.8s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 3\right) \cdot y - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * 3.0d0) * y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Python

def code(x, y, z):
	return ((x * 3.0) * y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * 3.0) * y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 3\right) \cdot y - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * 3.0d0) * y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}

Python

def code(x, y, z):
	return ((x * 3.0) * y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * 3.0) * y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * 3.0) * y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(x \cdot y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* 3.0 (* x y)) z))

C

double code(double x, double y, double z) {
	return (3.0 * (x * y)) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (3.0d0 * (x * y)) - z
end function

Java

public static double code(double x, double y, double z) {
	return (3.0 * (x * y)) - z;
}

Python

def code(x, y, z):
	return (3.0 * (x * y)) - z

Julia

function code(x, y, z)
	return Float64(Float64(3.0 * Float64(x * y)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (3.0 * (x * y)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} - z \]

6. Final simplification 99.8%

   \[\leadsto 3 \cdot \left(x \cdot y\right) - z \]
7. Add Preprocessing

Alternative 2: 68.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-141} \lor \neg \left(y \leq 2.35 \cdot 10^{-20}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.75e-141) (not (<= y 2.35e-20))) (* 3.0 (* x y)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.75e-141) || !(y <= 2.35e-20)) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.75d-141)) .or. (.not. (y <= 2.35d-20))) then
        tmp = 3.0d0 * (x * y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.75e-141) || !(y <= 2.35e-20)) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.75e-141) or not (y <= 2.35e-20):
		tmp = 3.0 * (x * y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.75e-141) || !(y <= 2.35e-20))
		tmp = Float64(3.0 * Float64(x * y));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.75e-141) || ~((y <= 2.35e-20)))
		tmp = 3.0 * (x * y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.75e-141], N[Not[LessEqual[y, 2.35e-20]], $MachinePrecision]], N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if y < -1.7500000000000001e-141 or 2.35000000000000007e-20 < y

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Step-by-step derivation

      1. associate-*l* 99.7%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} - z \]

   6. Taylor expanded in x around inf 70.3%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]

   if -1.7500000000000001e-141 < y < 2.35000000000000007e-20

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Step-by-step derivation

      1. associate-*l* 99.9%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 80.8%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   6. Step-by-step derivation

      1. neg-mul-1 80.8%

         \[\leadsto \color{blue}{-z} \]

   7. Simplified 80.8%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-141} \lor \neg \left(y \leq 2.35 \cdot 10^{-20}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-141)
   (* y (* 3.0 x))
   (if (<= y 2.65e-20) (- z) (* 3.0 (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-141) {
		tmp = y * (3.0 * x);
	} else if (y <= 2.65e-20) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2d-141)) then
        tmp = y * (3.0d0 * x)
    else if (y <= 2.65d-20) then
        tmp = -z
    else
        tmp = 3.0d0 * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-141) {
		tmp = y * (3.0 * x);
	} else if (y <= 2.65e-20) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-141:
		tmp = y * (3.0 * x)
	elif y <= 2.65e-20:
		tmp = -z
	else:
		tmp = 3.0 * (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-141)
		tmp = Float64(y * Float64(3.0 * x));
	elseif (y <= 2.65e-20)
		tmp = Float64(-z);
	else
		tmp = Float64(3.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2e-141)
		tmp = y * (3.0 * x);
	elseif (y <= 2.65e-20)
		tmp = -z;
	else
		tmp = 3.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-141], N[(y * N[(3.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.65e-20], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0000000000000001e-141

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Step-by-step derivation

      1. associate-*l* 99.7%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} - z \]

   6. Taylor expanded in x around inf 66.0%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]
   7. Step-by-step derivation

      1. *-commutative 66.0%

         \[\leadsto 3 \cdot \color{blue}{\left(y \cdot x\right)} \]

      2. associate-*l* 66.0%

         \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot x} \]

      3. *-commutative 66.0%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} \]

   8. Simplified 66.0%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} \]

   9. Taylor expanded in x around 0 66.0%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-commutative 66.0%

          \[\leadsto 3 \cdot \color{blue}{\left(y \cdot x\right)} \]

       2. associate-*r* 66.0%

          \[\leadsto \color{blue}{\left(3 \cdot y\right) \cdot x} \]

       3. *-commutative 66.0%

          \[\leadsto \color{blue}{\left(y \cdot 3\right)} \cdot x \]

       4. associate-*r* 66.1%

          \[\leadsto \color{blue}{y \cdot \left(3 \cdot x\right)} \]

   11. Simplified 66.1%

       \[\leadsto \color{blue}{y \cdot \left(3 \cdot x\right)} \]

   if -2.0000000000000001e-141 < y < 2.6500000000000001e-20

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Step-by-step derivation

      1. associate-*l* 99.9%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 80.8%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   6. Step-by-step derivation

      1. neg-mul-1 80.8%

         \[\leadsto \color{blue}{-z} \]

   7. Simplified 80.8%

      \[\leadsto \color{blue}{-z} \]

   if 2.6500000000000001e-20 < y

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Step-by-step derivation

      1. associate-*l* 99.7%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.9%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} - z \]

   6. Taylor expanded in x around inf 76.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

double code(double x, double y, double z) {
	return -z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -z
end function

Java

public static double code(double x, double y, double z) {
	return -z;
}

Python

def code(x, y, z):
	return -z

Julia

function code(x, y, z)
	return Float64(-z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -z;
end

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 48.4%

   \[\leadsto \color{blue}{-1 \cdot z} \]
6. Step-by-step derivation

   1. neg-mul-1 48.4%

      \[\leadsto \color{blue}{-z} \]

7. Simplified 48.4%

   \[\leadsto \color{blue}{-z} \]

8. Final simplification 48.4%

   \[\leadsto -z \]
9. Add Preprocessing

Alternative 5: 2.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 z)

C

double code(double x, double y, double z) {
	return z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = z
end function

Java

public static double code(double x, double y, double z) {
	return z;
}

Python

def code(x, y, z):
	return z

Julia

function code(x, y, z)
	return z
end

MATLAB

function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Step-by-step derivation

   1. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right)} - z \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{x \cdot \left(3 \cdot y\right) - z} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} - z \]
6. Step-by-step derivation

   1. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot 3} - z \]

   2. fma-neg 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot y, 3, -z\right)} \]

   3. add-sqr-sqrt 48.6%

      \[\leadsto \mathsf{fma}\left(x \cdot y, 3, \color{blue}{\sqrt{-z} \cdot \sqrt{-z}}\right) \]

   4. sqrt-unprod 58.0%

      \[\leadsto \mathsf{fma}\left(x \cdot y, 3, \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}}\right) \]

   5. sqr-neg 58.0%

      \[\leadsto \mathsf{fma}\left(x \cdot y, 3, \sqrt{\color{blue}{z \cdot z}}\right) \]

   6. sqrt-unprod 25.6%

      \[\leadsto \mathsf{fma}\left(x \cdot y, 3, \color{blue}{\sqrt{z} \cdot \sqrt{z}}\right) \]

   7. add-sqr-sqrt 52.7%

      \[\leadsto \mathsf{fma}\left(x \cdot y, 3, \color{blue}{z}\right) \]

7. Applied egg-rr 52.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot y, 3, z\right)} \]

8. Taylor expanded in x around 0 2.3%

   \[\leadsto \color{blue}{z} \]

9. Final simplification 2.3%

   \[\leadsto z \]
10. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(3 \cdot y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (* 3.0 y)) z))

C

double code(double x, double y, double z) {
	return (x * (3.0 * y)) - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (3.0d0 * y)) - z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (3.0 * y)) - z;
}

Python

def code(x, y, z):
	return (x * (3.0 * y)) - z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(3.0 * y)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (3.0 * y)) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Reproduce

herbie shell --seed 2024039 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))