Linear.Quaternion:$c/ from linear-1.19.1.3, B

Percentage Accurate: 62.0% → 100.0%

Time: 6.2s

Alternatives: 3

Speedup: 3.3×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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 * y) - (y * z)) - (y * y)) + (y * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * y) - (y * z)) - (y * y)) + (y * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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 - z) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-+l- N/A

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

   7. +-inverses N/A

      \[\leadsto \left(x \cdot y - y \cdot z\right) - \color{blue}{0} \]

   8. --rgt-identity 95.7

      \[\leadsto \color{blue}{x \cdot y - y \cdot z} \]

   9. lift--.f64 N/A

      \[\leadsto \color{blue}{x \cdot y - y \cdot z} \]

   10. lift-*.f64 N/A

       \[\leadsto x \cdot y - \color{blue}{y \cdot z} \]

   11. lift-*.f64 N/A

       \[\leadsto \color{blue}{x \cdot y} - y \cdot z \]

   12. *-commutative N/A

       \[\leadsto \color{blue}{y \cdot x} - y \cdot z \]

   13. distribute-lft-out-- N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 78.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -z \cdot y\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+53}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* z y))))
   (if (<= z -1.02e-44) t_0 (if (<= z 1.1e+53) (* x y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -(z * y);
	double tmp;
	if (z <= -1.02e-44) {
		tmp = t_0;
	} else if (z <= 1.1e+53) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -(z * y)
    if (z <= (-1.02d-44)) then
        tmp = t_0
    else if (z <= 1.1d+53) then
        tmp = x * y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -(z * y);
	double tmp;
	if (z <= -1.02e-44) {
		tmp = t_0;
	} else if (z <= 1.1e+53) {
		tmp = x * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -(z * y)
	tmp = 0
	if z <= -1.02e-44:
		tmp = t_0
	elif z <= 1.1e+53:
		tmp = x * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-Float64(z * y))
	tmp = 0.0
	if (z <= -1.02e-44)
		tmp = t_0;
	elseif (z <= 1.1e+53)
		tmp = Float64(x * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -(z * y);
	tmp = 0.0;
	if (z <= -1.02e-44)
		tmp = t_0;
	elseif (z <= 1.1e+53)
		tmp = x * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = (-N[(z * y), $MachinePrecision])}, If[LessEqual[z, -1.02e-44], t$95$0, If[LessEqual[z, 1.1e+53], N[(x * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0199999999999999e-44 or 1.09999999999999999e53 < z

   1. Initial program 68.8%

      \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y \cdot z\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{y \cdot \left(\mathsf{neg}\left(z\right)\right)} \]

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto y \cdot \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      6. lower-neg.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -1.0199999999999999e-44 < z < 1.09999999999999999e53

   1. Initial program 55.0%

      \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot x} \]

      2. lower-*.f64 82.6

         \[\leadsto \color{blue}{y \cdot x} \]

   5. Applied rewrites 82.6%

      \[\leadsto \color{blue}{y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{-44}:\\ \;\;\;\;-z \cdot y\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+53}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

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

C

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

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 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

   \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot y} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot x} \]

   2. lower-*.f64 51.6

      \[\leadsto \color{blue}{y \cdot x} \]

5. Applied rewrites 51.6%

   \[\leadsto \color{blue}{y \cdot x} \]

6. Final simplification 51.6%

   \[\leadsto x \cdot y \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

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 - z) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024216 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, B"
  :precision binary64

  :alt
  (! :herbie-platform default (* (- x z) y))

  (+ (- (- (* x y) (* y z)) (* y y)) (* y y)))