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

Percentage Accurate: 99.8% → 99.9%

Time: 2.4s

Alternatives: 3

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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \mathsf{fma}\left(y, x, x \cdot \left(y \cdot 2\right)\right) - z \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- (fma y x (* x (* y 2.0))) z))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return fma(y, x, (x * (y * 2.0))) - z;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(fma(y, x, Float64(x * Float64(y * 2.0))) - z)
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(y * x + N[(x * N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.1%

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

3. Simplified 99.1%

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

5. Taylor expanded in x around 0 99.8%

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

   1. add-log-exp 56.7%

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

   2. add-cube-cbrt 56.7%

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

   3. log-prod 56.7%

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

   4. pow2 56.7%

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

   5. *-commutative 56.7%

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

   6. exp-prod 56.7%

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

   7. pow3 56.7%

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

   8. add-cbrt-cube 56.7%

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

   9. exp-prod 44.8%

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

   10. *-commutative 44.8%

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

   11. exp-prod 44.8%

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

   12. pow3 44.8%

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

   13. add-cbrt-cube 44.8%

       \[\leadsto \left(\log \left({\left({\left(e^{x}\right)}^{y}\right)}^{2}\right) + \log \color{blue}{\left(e^{x \cdot y}\right)}\right) - z \]

7. Applied egg-rr 51.3%

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

   1. +-commutative 51.3%

      \[\leadsto \color{blue}{\left(x \cdot y + \log \left({\left({\left(e^{x}\right)}^{y}\right)}^{2}\right)\right)} - z \]

   2. *-commutative 51.3%

      \[\leadsto \left(\color{blue}{y \cdot x} + \log \left({\left({\left(e^{x}\right)}^{y}\right)}^{2}\right)\right) - z \]

   3. fma-define 51.3%

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

   4. unpow2 51.3%

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

   5. exp-prod 45.5%

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

   6. exp-prod 58.0%

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

   7. prod-exp 58.1%

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

   8. rem-log-exp 99.9%

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

   9. distribute-lft-out 99.6%

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

   10. count-2 99.6%

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

   11. *-commutative 99.6%

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

9. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, x \cdot \left(y \cdot 2\right)\right)} - z \]
10. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- (* x (* y 3.0)) z))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (x * (y * 3.0)) - z;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (y * 3.0d0)) - z
end function

Java

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return (x * (y * 3.0)) - z

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(Float64(x * Float64(y * 3.0)) - z)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = (x * (y * 3.0)) - z;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(x * N[(y * 3.0), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (- (* 3.0 (* y x)) z))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (3.0 * (y * x)) - z;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return (3.0 * (y * x)) - z

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(Float64(3.0 * Float64(y * x)) - z)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = (3.0 * (y * x)) - z;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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. Step-by-step derivation

   1. associate-*l* 99.1%

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

3. Simplified 99.1%

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

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 2024106 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))