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

Percentage Accurate: 99.8% → 99.8%

Time: 5.0s

Alternatives: 4

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, 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. 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: 70.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1100000 \lor \neg \left(y \leq 6.5 \cdot 10^{+69}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1100000.0) (not (<= y 6.5e+69))) (* 3.0 (* y x)) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1100000.0) || !(y <= 6.5e+69)) {
		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 ((y <= (-1100000.0d0)) .or. (.not. (y <= 6.5d+69))) 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 ((y <= -1100000.0) || !(y <= 6.5e+69)) {
		tmp = 3.0 * (y * x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1100000.0) or not (y <= 6.5e+69):
		tmp = 3.0 * (y * x)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1100000.0) || !(y <= 6.5e+69))
		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 ((y <= -1100000.0) || ~((y <= 6.5e+69)))
		tmp = 3.0 * (y * x);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1100000.0], N[Not[LessEqual[y, 6.5e+69]], $MachinePrecision]], N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e6 or 6.5000000000000001e69 < y

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 43.7%

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

      2. associate-*r* 43.7%

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

      3. fma-neg 43.7%

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

      4. add-sqr-sqrt 25.7%

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

      5. sqrt-unprod 32.0%

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

      6. sqr-neg 32.0%

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

      7. sqrt-unprod 12.8%

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

      8. add-sqr-sqrt 32.1%

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

   3. Applied egg-rr 32.1%

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

      1. fma-udef 32.1%

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

      2. flip-+ 11.4%

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

   5. Applied egg-rr 26.1%

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

      1. fma-neg 26.1%

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

   7. Simplified 26.1%

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

      1. clear-num 26.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot \left(y \cdot x\right) - z}{{\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z}}} \]

      2. associate-/r/ 26.1%

         \[\leadsto \color{blue}{\frac{1}{3 \cdot \left(y \cdot x\right) - z} \cdot \left({\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z\right)} \]

      3. associate-*r* 26.0%

         \[\leadsto \frac{1}{\color{blue}{\left(3 \cdot y\right) \cdot x} - z} \cdot \left({\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z\right) \]

      4. *-commutative 26.0%

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(3 \cdot y\right)} - z} \cdot \left({\left(y \cdot x\right)}^{2} \cdot 9 - z \cdot z\right) \]

      5. *-commutative 26.0%

         \[\leadsto \frac{1}{x \cdot \left(3 \cdot y\right) - z} \cdot \left(\color{blue}{9 \cdot {\left(y \cdot x\right)}^{2}} - z \cdot z\right) \]

      6. metadata-eval 26.0%

         \[\leadsto \frac{1}{x \cdot \left(3 \cdot y\right) - z} \cdot \left(\color{blue}{\left(3 \cdot 3\right)} \cdot {\left(y \cdot x\right)}^{2} - z \cdot z\right) \]

      7. unpow2 26.0%

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

      8. swap-sqr 26.0%

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

      9. pow2 26.0%

         \[\leadsto \frac{1}{x \cdot \left(3 \cdot y\right) - z} \cdot \left(\color{blue}{{\left(3 \cdot \left(y \cdot x\right)\right)}^{2}} - z \cdot z\right) \]

      10. associate-*r* 26.1%

          \[\leadsto \frac{1}{x \cdot \left(3 \cdot y\right) - z} \cdot \left({\color{blue}{\left(\left(3 \cdot y\right) \cdot x\right)}}^{2} - z \cdot z\right) \]

      11. *-commutative 26.1%

          \[\leadsto \frac{1}{x \cdot \left(3 \cdot y\right) - z} \cdot \left({\color{blue}{\left(x \cdot \left(3 \cdot y\right)\right)}}^{2} - z \cdot z\right) \]

   9. Applied egg-rr 26.1%

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(3 \cdot y\right) - z} \cdot \left({\left(x \cdot \left(3 \cdot y\right)\right)}^{2} - z \cdot z\right)} \]

   10. Taylor expanded in x around inf 73.3%

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

   if -1.1e6 < y < 6.5000000000000001e69

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 74.4%

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

      1. mul-1-neg 74.4%

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

   4. Simplified 74.4%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1100000 \lor \neg \left(y \leq 6.5 \cdot 10^{+69}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 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 99.8%

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

2. Taylor expanded in x around 0 52.5%

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

   1. mul-1-neg 52.5%

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

4. Simplified 52.5%

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

5. Final simplification 52.5%

   \[\leadsto -z \]

Alternative 4: 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. add-sqr-sqrt 46.3%

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

   2. associate-*r* 46.3%

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

   3. fma-neg 46.3%

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

   4. add-sqr-sqrt 25.2%

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

   5. sqrt-unprod 29.6%

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

   6. sqr-neg 29.6%

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

   7. sqrt-unprod 10.7%

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

   8. add-sqr-sqrt 22.9%

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

3. Applied egg-rr 22.9%

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

4. Taylor expanded in x around 0 2.4%

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

5. Final simplification 2.4%

   \[\leadsto z \]

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 2023229 
(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))