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

Percentage Accurate: 62.9% → 100.0%

Time: 5.2s

Alternatives: 3

Speedup: 3.0×

• 21.2% 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: 62.9% 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, 3.0× 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 61.7%

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

   1. +-commutative N/A

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

   2. associate--l- N/A

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

   3. associate-+r- N/A

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

   4. distribute-rgt-out N/A

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

   5. distribute-lft-out N/A

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

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

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

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(y + x\right) - \left(z + y\right)\right)}\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y + x\right) - \left(y + \color{blue}{z}\right)\right)\right) \]

   9. associate--r+ N/A

      \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\left(y + x\right) - y\right) - \color{blue}{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\left(x + y\right) - y\right) - z\right)\right) \]

   11. associate--l+ N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\left(x + \left(y - y\right)\right) - z\right)\right) \]

   12. +-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\left(x + 0\right) - z\right)\right) \]

   13. +-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(x - z\right)\right) \]

   14. --lowering--.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(x, \color{blue}{z}\right)\right) \]

3. Simplified 100.0%

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

Alternative 2: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(0 - z\right)\\ \mathbf{if}\;z \leq -1.36 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-28}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (- 0.0 z))))
   (if (<= z -1.36e-17) t_0 (if (<= z 3.3e-28) (* y x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y * (0.0 - z);
	double tmp;
	if (z <= -1.36e-17) {
		tmp = t_0;
	} else if (z <= 3.3e-28) {
		tmp = y * x;
	} 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 = y * (0.0d0 - z)
    if (z <= (-1.36d-17)) then
        tmp = t_0
    else if (z <= 3.3d-28) then
        tmp = y * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (0.0 - z);
	double tmp;
	if (z <= -1.36e-17) {
		tmp = t_0;
	} else if (z <= 3.3e-28) {
		tmp = y * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (0.0 - z)
	tmp = 0
	if z <= -1.36e-17:
		tmp = t_0
	elif z <= 3.3e-28:
		tmp = y * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(0.0 - z))
	tmp = 0.0
	if (z <= -1.36e-17)
		tmp = t_0;
	elseif (z <= 3.3e-28)
		tmp = Float64(y * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (0.0 - z);
	tmp = 0.0;
	if (z <= -1.36e-17)
		tmp = t_0;
	elseif (z <= 3.3e-28)
		tmp = y * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.36e-17], t$95$0, If[LessEqual[z, 3.3e-28], N[(y * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.36e-17 or 3.3000000000000002e-28 < z

   1. Initial program 75.4%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. distribute-rgt-out N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(y + x\right) - \left(z + y\right)\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y + x\right) - \left(y + \color{blue}{z}\right)\right)\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\left(y + x\right) - y\right) - \color{blue}{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\left(x + y\right) - y\right) - z\right)\right) \]

      11. associate--l+ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(x + \left(y - y\right)\right) - z\right)\right) \]

      12. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(x + 0\right) - z\right)\right) \]

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x - z\right)\right) \]

      14. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(x, \color{blue}{z}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{y \cdot z} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(y \cdot z\right)}\right) \]

      4. *-lowering-*.f64 76.6%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 76.6%

      \[\leadsto \color{blue}{0 - y \cdot z} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

      2. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\left(0 + y \cdot z\right)\right) \]

      3. flip3-+ N/A

         \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(y \cdot z\right)}^{3}}{0 \cdot 0 + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - 0 \cdot \left(y \cdot z\right)\right)}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\frac{0 + {\left(y \cdot z\right)}^{3}}{0 \cdot 0 + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - 0 \cdot \left(y \cdot z\right)\right)}\right) \]

      5. +-lft-identity N/A

         \[\leadsto \mathsf{neg}\left(\frac{{\left(y \cdot z\right)}^{3}}{0 \cdot 0 + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - 0 \cdot \left(y \cdot z\right)\right)}\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \frac{\mathsf{neg}\left({\left(y \cdot z\right)}^{3}\right)}{\color{blue}{0 \cdot 0 + \left(\left(y \cdot z\right) \cdot \left(y \cdot z\right) - 0 \cdot \left(y \cdot z\right)\right)}} \]

   9. Applied egg-rr 76.6%

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

   if -1.36e-17 < z < 3.3000000000000002e-28

   1. Initial program 44.6%

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

      1. +-commutative N/A

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

      2. associate--l- N/A

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

      3. associate-+r- N/A

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

      4. distribute-rgt-out N/A

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

      5. distribute-lft-out N/A

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

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

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(y + x\right) - \left(z + y\right)\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y + x\right) - \left(y + \color{blue}{z}\right)\right)\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\left(y + x\right) - y\right) - \color{blue}{z}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\left(x + y\right) - y\right) - z\right)\right) \]

      11. associate--l+ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(x + \left(y - y\right)\right) - z\right)\right) \]

      12. +-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(\left(x + 0\right) - z\right)\right) \]

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(y, \left(x - z\right)\right) \]

      14. --lowering--.f64 100.0%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(x, \color{blue}{z}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

   7. Simplified 83.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{-17}:\\ \;\;\;\;y \cdot \left(0 - z\right)\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-28}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.8% 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 61.7%

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

   1. +-commutative N/A

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

   2. associate--l- N/A

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

   3. associate-+r- N/A

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

   4. distribute-rgt-out N/A

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

   5. distribute-lft-out N/A

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

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

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

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(\left(y + x\right) - \left(z + y\right)\right)}\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(y, \left(\left(y + x\right) - \left(y + \color{blue}{z}\right)\right)\right) \]

   9. associate--r+ N/A

      \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\left(y + x\right) - y\right) - \color{blue}{z}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\left(\left(x + y\right) - y\right) - z\right)\right) \]

   11. associate--l+ N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\left(x + \left(y - y\right)\right) - z\right)\right) \]

   12. +-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(\left(x + 0\right) - z\right)\right) \]

   13. +-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(y, \left(x - z\right)\right) \]

   14. --lowering--.f64 100.0%

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(x, \color{blue}{z}\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 53.4%

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{x}\right) \]

7. Simplified 53.4%

   \[\leadsto \color{blue}{y \cdot x} \]
8. Add Preprocessing

Developer Target 1: 100.0% accurate, 3.0× 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 2024145 
(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)))