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

Percentage Accurate: 99.8% → 99.8%

Time: 4.6s

Alternatives: 7

Speedup: N/A×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. *-commutative 99.8%

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

   3. fma-neg 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+59} \lor \neg \left(x \leq 9.8 \cdot 10^{-107}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.8e+59) (not (<= x 9.8e-107))) (* 3.0 (* y x)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.8e+59) || !(x <= 9.8e-107)) {
		tmp = 3.0 * (y * x);
	} 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 ((x <= (-3.8d+59)) .or. (.not. (x <= 9.8d-107))) then
        tmp = 3.0d0 * (y * x)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.8e+59) || !(x <= 9.8e-107)) {
		tmp = 3.0 * (y * x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.8e+59) or not (x <= 9.8e-107):
		tmp = 3.0 * (y * x)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.8e+59) || !(x <= 9.8e-107))
		tmp = Float64(3.0 * Float64(y * x));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.8e+59) || ~((x <= 9.8e-107)))
		tmp = 3.0 * (y * x);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.8e+59], N[Not[LessEqual[x, 9.8e-107]], $MachinePrecision]], N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000001e59 or 9.79999999999999959e-107 < x

   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. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in x around inf 71.8%

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

   if -3.8000000000000001e59 < x < 9.79999999999999959e-107

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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. Taylor expanded in x around 0 73.1%

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

      1. mul-1-neg 73.1%

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

   6. Simplified 73.1%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+59} \lor \neg \left(x \leq 9.8 \cdot 10^{-107}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+48}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-104}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.25e+48)
   (* (* 3.0 y) x)
   (if (<= x 1.4e-104) (- z) (* 3.0 (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e+48) {
		tmp = (3.0 * y) * x;
	} else if (x <= 1.4e-104) {
		tmp = -z;
	} else {
		tmp = 3.0 * (y * x);
	}
	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 (x <= (-1.25d+48)) then
        tmp = (3.0d0 * y) * x
    else if (x <= 1.4d-104) then
        tmp = -z
    else
        tmp = 3.0d0 * (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.25e+48) {
		tmp = (3.0 * y) * x;
	} else if (x <= 1.4e-104) {
		tmp = -z;
	} else {
		tmp = 3.0 * (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.25e+48:
		tmp = (3.0 * y) * x
	elif x <= 1.4e-104:
		tmp = -z
	else:
		tmp = 3.0 * (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.25e+48)
		tmp = Float64(Float64(3.0 * y) * x);
	elseif (x <= 1.4e-104)
		tmp = Float64(-z);
	else
		tmp = Float64(3.0 * Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.25e+48)
		tmp = (3.0 * y) * x;
	elseif (x <= 1.4e-104)
		tmp = -z;
	else
		tmp = 3.0 * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.25e+48], N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.4e-104], (-z), N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.24999999999999993e48

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in x around inf 72.5%

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

      1. associate-*r* 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-*r* 72.6%

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

   7. Simplified 72.6%

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

   if -1.24999999999999993e48 < x < 1.4e-104

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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. Taylor expanded in x around 0 74.9%

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

      1. mul-1-neg 74.9%

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

   6. Simplified 74.9%

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

   if 1.4e-104 < x

   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. Taylor expanded in x around 0 99.8%

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

   5. Taylor expanded in x around inf 71.5%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+48}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-104}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(3.0 * N[(y * x), $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. Taylor expanded in x around 0 99.8%

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

5. Final simplification 99.8%

   \[\leadsto 3 \cdot \left(y \cdot x\right) - z \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(3.0 * y), $MachinePrecision] * x), $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. Final simplification 99.8%

   \[\leadsto \left(3 \cdot y\right) \cdot x - z \]

Alternative 6: 50.5% 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. Taylor expanded in x around 0 48.1%

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

   1. mul-1-neg 48.1%

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

6. Simplified 48.1%

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

7. Final simplification 48.1%

   \[\leadsto -z \]

Alternative 7: 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. Step-by-step derivation

   1. add-sqr-sqrt 51.8%

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

   2. associate-*r* 51.8%

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

   3. fma-neg 51.8%

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

   4. add-sqr-sqrt 25.2%

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

   5. sqrt-unprod 31.8%

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

   6. sqr-neg 31.8%

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

   7. sqrt-unprod 16.0%

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

   8. add-sqr-sqrt 27.2%

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

5. Applied egg-rr 27.2%

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

6. Taylor expanded in x around 0 2.1%

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

7. Final simplification 2.1%

   \[\leadsto z \]

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 2023334 
(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))