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

Percentage Accurate: 99.8% → 99.8%

Time: 8.1s

Alternatives: 7

Speedup: 0.1×

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* 99.9%

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

   2. *-commutative 99.9%

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

   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. Add Preprocessing

5. Final simplification 99.9%

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

Alternative 2: 68.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-103} \lor \neg \left(z \leq 2.45 \cdot 10^{+89}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.1e-103) (not (<= z 2.45e+89))) (- z) (* 3.0 (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.1e-103) || !(z <= 2.45e+89)) {
		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 ((z <= (-3.1d-103)) .or. (.not. (z <= 2.45d+89))) 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 ((z <= -3.1e-103) || !(z <= 2.45e+89)) {
		tmp = -z;
	} else {
		tmp = 3.0 * (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.1e-103) or not (z <= 2.45e+89):
		tmp = -z
	else:
		tmp = 3.0 * (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.1e-103) || !(z <= 2.45e+89))
		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 ((z <= -3.1e-103) || ~((z <= 2.45e+89)))
		tmp = -z;
	else
		tmp = 3.0 * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.1e-103], N[Not[LessEqual[z, 2.45e+89]], $MachinePrecision]], (-z), N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1000000000000001e-103 or 2.44999999999999998e89 < z

   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 75.8%

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

      1. mul-1-neg 75.8%

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

   7. Simplified 75.8%

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

   if -3.1000000000000001e-103 < z < 2.44999999999999998e89

   1. Initial program 99.7%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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 99.7%

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

   6. Taylor expanded in x around inf 77.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-103} \lor \neg \left(z \leq 2.45 \cdot 10^{+89}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-103} \lor \neg \left(z \leq 6.5 \cdot 10^{+93}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.1e-103) (not (<= z 6.5e+93))) (- z) (* (* 3.0 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.1e-103) || !(z <= 6.5e+93)) {
		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 ((z <= (-3.1d-103)) .or. (.not. (z <= 6.5d+93))) 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 ((z <= -3.1e-103) || !(z <= 6.5e+93)) {
		tmp = -z;
	} else {
		tmp = (3.0 * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.1e-103) or not (z <= 6.5e+93):
		tmp = -z
	else:
		tmp = (3.0 * y) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.1e-103) || !(z <= 6.5e+93))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(3.0 * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -3.1e-103) || ~((z <= 6.5e+93)))
		tmp = -z;
	else
		tmp = (3.0 * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -3.1e-103], N[Not[LessEqual[z, 6.5e+93]], $MachinePrecision]], (-z), N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.1000000000000001e-103 or 6.4999999999999998e93 < z

   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 75.8%

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

      1. mul-1-neg 75.8%

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

   7. Simplified 75.8%

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

   if -3.1000000000000001e-103 < z < 6.4999999999999998e93

   1. Initial program 99.7%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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 99.7%

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

   6. Taylor expanded in x around inf 77.0%

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

      1. associate-*r* 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-*r* 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-103} \lor \neg \left(z \leq 6.5 \cdot 10^{+93}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

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* 99.9%

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

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 99.8%

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

6. Final simplification 99.8%

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

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* 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 51.0% 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* 99.9%

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

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 51.1%

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

   1. mul-1-neg 51.1%

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

7. Simplified 51.1%

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

8. Final simplification 51.1%

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

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* 99.9%

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

3. Simplified 99.9%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*r* 99.8%

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

   4. fma-neg 99.8%

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

   5. add-sqr-sqrt 50.1%

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

   6. sqrt-unprod 61.5%

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

   7. sqr-neg 61.5%

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

   8. sqrt-unprod 26.7%

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

   9. add-sqr-sqrt 48.7%

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

6. Applied egg-rr 48.7%

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

7. Taylor expanded in y around 0 2.4%

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

8. Final simplification 2.4%

   \[\leadsto z \]
9. 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 2024031 
(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))