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

Percentage Accurate: 99.8% → 99.8%

Time: 4.4s

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

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

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]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+58} \lor \neg \left(x \leq 2.9 \cdot 10^{-49}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.8e+58) (not (<= x 2.9e-49))) (* 3.0 (* x y)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.8e+58) || !(x <= 2.9e-49)) {
		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 ((x <= (-6.8d+58)) .or. (.not. (x <= 2.9d-49))) 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 ((x <= -6.8e+58) || !(x <= 2.9e-49)) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -6.8e+58) or not (x <= 2.9e-49):
		tmp = 3.0 * (x * y)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.8e+58) || !(x <= 2.9e-49))
		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 ((x <= -6.8e+58) || ~((x <= 2.9e-49)))
		tmp = 3.0 * (x * y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.8e+58], N[Not[LessEqual[x, 2.9e-49]], $MachinePrecision]], N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -6.8000000000000001e58 or 2.9e-49 < x

   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. associate-*r* 99.8%

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

      3. fma-neg 99.8%

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

      4. add-sqr-sqrt 43.9%

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

      5. sqrt-unprod 67.9%

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

      6. sqr-neg 67.9%

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

      7. sqrt-unprod 36.0%

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

      8. add-sqr-sqrt 65.8%

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

   3. Applied egg-rr 65.8%

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

      1. fma-udef 65.8%

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

      2. *-commutative 65.8%

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

      3. associate-*r* 65.7%

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

      4. *-commutative 65.7%

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

      5. flip-+ 21.6%

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

      6. *-commutative 21.6%

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

      7. associate-*r* 21.7%

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

      8. *-commutative 21.7%

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

      9. *-commutative 21.7%

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

      10. associate-*r* 21.6%

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

      11. *-commutative 21.6%

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

      12. swap-sqr 21.5%

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

      13. pow2 21.5%

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

      14. metadata-eval 21.5%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot \color{blue}{9} - z \cdot z}{x \cdot \left(3 \cdot y\right) - z} \]

      15. *-commutative 21.5%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\color{blue}{\left(3 \cdot y\right) \cdot x} - z} \]

      16. associate-*r* 21.6%

          \[\leadsto \frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{\color{blue}{3 \cdot \left(y \cdot x\right)} - z} \]

   5. Applied egg-rr 21.6%

      \[\leadsto \color{blue}{\frac{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}{3 \cdot \left(y \cdot x\right) - z}} \]
   6. Step-by-step derivation

      1. unpow2 21.6%

         \[\leadsto \frac{\color{blue}{\left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right)} \cdot 9 - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

      2. *-commutative 21.6%

         \[\leadsto \frac{\left(\left(y \cdot x\right) \cdot \color{blue}{\left(x \cdot y\right)}\right) \cdot 9 - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

      3. associate-*r* 18.1%

         \[\leadsto \frac{\color{blue}{\left(\left(\left(y \cdot x\right) \cdot x\right) \cdot y\right)} \cdot 9 - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

      4. *-commutative 18.1%

         \[\leadsto \frac{\left(\left(\color{blue}{\left(x \cdot y\right)} \cdot x\right) \cdot y\right) \cdot 9 - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

   7. Applied egg-rr 18.1%

      \[\leadsto \frac{\color{blue}{\left(\left(\left(x \cdot y\right) \cdot x\right) \cdot y\right)} \cdot 9 - z \cdot z}{3 \cdot \left(y \cdot x\right) - z} \]

   8. Taylor expanded in x around inf 66.7%

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

   if -6.8000000000000001e58 < x < 2.9e-49

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 72.3%

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

      1. mul-1-neg 72.3%

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

   4. Simplified 72.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+58} \lor \neg \left(x \leq 2.9 \cdot 10^{-49}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 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 99.9%

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

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%

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

Alternative 4: 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 99.9%

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

2. Taylor expanded in x around 0 52.7%

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

   1. mul-1-neg 52.7%

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

4. Simplified 52.7%

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

5. Final simplification 52.7%

   \[\leadsto -z \]

Alternative 5: 2.2% 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. associate-*r* 99.8%

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

   3. fma-neg 99.9%

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

   4. add-sqr-sqrt 45.9%

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

   5. sqrt-unprod 58.0%

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

   6. sqr-neg 58.0%

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

   7. sqrt-unprod 26.4%

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

   8. add-sqr-sqrt 47.5%

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

3. Applied egg-rr 47.5%

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

4. Taylor expanded in y around 0 2.3%

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

5. Final simplification 2.3%

   \[\leadsto z \]

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 2023228 
(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))