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

Percentage Accurate: 99.8% → 99.8%

Time: 4.3s

Alternatives: 6

Speedup: 1.0×

• Unsound rule application detected in e-graph. Results may not be sound.

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 99.9%

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

   1. *-commutative 99.9%

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

   2. fma-neg 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. fma-neg 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 67.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-138} \lor \neg \left(y \leq 300000000\right) \land \left(y \leq 2.55 \cdot 10^{+72} \lor \neg \left(y \leq 1.75 \cdot 10^{+115}\right)\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.5e-138)
         (and (not (<= y 300000000.0))
              (or (<= y 2.55e+72) (not (<= y 1.75e+115)))))
   (* (* y x) 3.0)
   (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.5e-138) || (!(y <= 300000000.0) && ((y <= 2.55e+72) || !(y <= 1.75e+115)))) {
		tmp = (y * x) * 3.0;
	} 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 <= (-1.5d-138)) .or. (.not. (y <= 300000000.0d0)) .and. (y <= 2.55d+72) .or. (.not. (y <= 1.75d+115))) then
        tmp = (y * x) * 3.0d0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.5e-138) || (!(y <= 300000000.0) && ((y <= 2.55e+72) || !(y <= 1.75e+115)))) {
		tmp = (y * x) * 3.0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.5e-138) or (not (y <= 300000000.0) and ((y <= 2.55e+72) or not (y <= 1.75e+115))):
		tmp = (y * x) * 3.0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.5e-138) || (!(y <= 300000000.0) && ((y <= 2.55e+72) || !(y <= 1.75e+115))))
		tmp = Float64(Float64(y * x) * 3.0);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.5e-138) || (~((y <= 300000000.0)) && ((y <= 2.55e+72) || ~((y <= 1.75e+115)))))
		tmp = (y * x) * 3.0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.5e-138], And[N[Not[LessEqual[y, 300000000.0]], $MachinePrecision], Or[LessEqual[y, 2.55e+72], N[Not[LessEqual[y, 1.75e+115]], $MachinePrecision]]]], N[(N[(y * x), $MachinePrecision] * 3.0), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if y < -1.5e-138 or 3e8 < y < 2.54999999999999989e72 or 1.75000000000000003e115 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

   3. Taylor expanded in y around inf 67.7%

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

   if -1.5e-138 < y < 3e8 or 2.54999999999999989e72 < y < 1.75000000000000003e115

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 72.8%

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

      1. mul-1-neg 72.8%

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

   4. Simplified 72.8%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-138} \lor \neg \left(y \leq 300000000\right) \land \left(y \leq 2.55 \cdot 10^{+72} \lor \neg \left(y \leq 1.75 \cdot 10^{+115}\right)\right):\\ \;\;\;\;\left(y \cdot x\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((y * x) * 3.0) - 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 = ((y * x) * 3.0d0) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 50.4% 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 53.6%

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

   1. mul-1-neg 53.6%

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

4. Simplified 53.6%

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

5. Final simplification 53.6%

   \[\leadsto -z \]

Alternative 6: 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 99.9%

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

   1. *-commutative 99.9%

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

   2. add-cube-cbrt 99.2%

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

   3. associate-*l* 99.2%

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

   4. fma-neg 99.2%

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

   5. add-sqr-sqrt 45.2%

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

   6. sqrt-unprod 57.7%

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

   7. sqr-neg 57.7%

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

   8. sqrt-unprod 27.3%

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

   9. add-sqr-sqrt 46.7%

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

   10. pow2 46.7%

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

3. Applied egg-rr 46.7%

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

4. Taylor expanded in y around 0 2.2%

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

5. Final simplification 2.2%

   \[\leadsto z \]

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 2023174 
(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))