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

Percentage Accurate: 62.0% → 99.1%

Time: 6.7s

Alternatives: 4

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: 99.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. +-inverses N/A

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

   5. --rgt-identity 97.2

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-neg.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 78.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-38}:\\ \;\;\;\;-z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.2e-5) (* y x) (if (<= x 4e-38) (- (* z y)) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e-5) {
		tmp = y * x;
	} else if (x <= 4e-38) {
		tmp = -(z * y);
	} else {
		tmp = y * x;
	}
	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 <= (-5.2d-5)) then
        tmp = y * x
    else if (x <= 4d-38) then
        tmp = -(z * y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e-5) {
		tmp = y * x;
	} else if (x <= 4e-38) {
		tmp = -(z * y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.2e-5:
		tmp = y * x
	elif x <= 4e-38:
		tmp = -(z * y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.2e-5)
		tmp = Float64(y * x);
	elseif (x <= 4e-38)
		tmp = Float64(-Float64(z * y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.2e-5)
		tmp = y * x;
	elseif (x <= 4e-38)
		tmp = -(z * y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.2e-5], N[(y * x), $MachinePrecision], If[LessEqual[x, 4e-38], (-N[(z * y), $MachinePrecision]), N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999968e-5 or 3.9999999999999998e-38 < x

   1. Initial program 62.1%

      \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
   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 86.0

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

   5. Applied rewrites 86.0%

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

   if -5.19999999999999968e-5 < x < 3.9999999999999998e-38

   1. Initial program 46.2%

      \[\left(\left(x \cdot y - y \cdot z\right) - y \cdot y\right) + y \cdot y \]
   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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 79.5

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

   5. Applied rewrites 79.5%

      \[\leadsto \color{blue}{z \cdot \left(-y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-5}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-38}:\\ \;\;\;\;-z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. +-inverses N/A

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

   5. --rgt-identity 97.2

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-out-- N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. 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. Final simplification 100.0%

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

Alternative 4: 54.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

3. Taylor expanded in x around 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 58.9

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

5. Applied rewrites 58.9%

   \[\leadsto \color{blue}{y \cdot x} \]
6. 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 2024233 
(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)))