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

Percentage Accurate: 99.8% → 99.8%

Time: 3.5s

Alternatives: 4

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

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

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 99.9%

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

2. Taylor expanded in x around 0 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+89} \lor \neg \left(x \leq 1.9 \cdot 10^{-139}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.35e+89) (not (<= x 1.9e-139))) (* 3.0 (* y x)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.35e+89) || !(x <= 1.9e-139)) {
		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 <= (-1.35d+89)) .or. (.not. (x <= 1.9d-139))) 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 <= -1.35e+89) || !(x <= 1.9e-139)) {
		tmp = 3.0 * (y * x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.35e+89) or not (x <= 1.9e-139):
		tmp = 3.0 * (y * x)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.35e+89) || !(x <= 1.9e-139))
		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 <= -1.35e+89) || ~((x <= 1.9e-139)))
		tmp = 3.0 * (y * x);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.35e+89], N[Not[LessEqual[x, 1.9e-139]], $MachinePrecision]], N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e89 or 1.90000000000000004e-139 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   3. Taylor expanded in y around inf 69.9%

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

   if -1.35e89 < x < 1.90000000000000004e-139

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 70.1%

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

      1. mul-1-neg 70.1%

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

   4. Simplified 70.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+89} \lor \neg \left(x \leq 1.9 \cdot 10^{-139}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 66.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.24 \cdot 10^{+89}:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-141}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.24e+89)
   (* y (* 3.0 x))
   (if (<= x 2.1e-141) (- z) (* 3.0 (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.24e+89) {
		tmp = y * (3.0 * x);
	} else if (x <= 2.1e-141) {
		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 <= (-1.24d+89)) then
        tmp = y * (3.0d0 * x)
    else if (x <= 2.1d-141) 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 <= -1.24e+89) {
		tmp = y * (3.0 * x);
	} else if (x <= 2.1e-141) {
		tmp = -z;
	} else {
		tmp = 3.0 * (y * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.24e+89:
		tmp = y * (3.0 * x)
	elif x <= 2.1e-141:
		tmp = -z
	else:
		tmp = 3.0 * (y * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.24e+89)
		tmp = Float64(y * Float64(3.0 * x));
	elseif (x <= 2.1e-141)
		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 <= -1.24e+89)
		tmp = y * (3.0 * x);
	elseif (x <= 2.1e-141)
		tmp = -z;
	else
		tmp = 3.0 * (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.24e+89], N[(y * N[(3.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e-141], (-z), N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2400000000000001e89

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. add-cube-cbrt 98.8%

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

      3. associate-*l* 98.8%

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

      4. fma-neg 98.8%

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

      5. pow2 98.8%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{y \cdot x}\right)}^{2}}, \sqrt[3]{y \cdot x} \cdot 3, -z\right) \]

   4. Applied egg-rr 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{y \cdot x}\right)}^{2}, \sqrt[3]{y \cdot x} \cdot 3, -z\right)} \]

   5. Taylor expanded in z around 0 78.6%

      \[\leadsto \color{blue}{3 \cdot \left({1}^{0.3333333333333333} \cdot \left(y \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. pow-base-1 78.6%

         \[\leadsto 3 \cdot \left(\color{blue}{1} \cdot \left(y \cdot x\right)\right) \]

      2. *-lft-identity 78.6%

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

      3. *-commutative 78.6%

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

      4. associate-*l* 78.7%

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

      5. *-commutative 78.7%

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

   7. Simplified 78.7%

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

   if -1.2400000000000001e89 < x < 2.0999999999999999e-141

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 70.1%

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

      1. mul-1-neg 70.1%

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

   4. Simplified 70.1%

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

   if 2.0999999999999999e-141 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   3. Taylor expanded in y around inf 65.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.24 \cdot 10^{+89}:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-141}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \]

Alternative 4: 52.7% 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 99.9%

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

2. Taylor expanded in x around 0 50.7%

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

   1. mul-1-neg 50.7%

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

4. Simplified 50.7%

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

5. Final simplification 50.7%

   \[\leadsto -z \]

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 2023185 
(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))