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

Percentage Accurate: 98.5% → 98.9%

Time: 4.0s

Alternatives: 6

Speedup: 1.0×

• 34.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma z (* z 3.0) (* x y)))

C

double code(double x, double y, double z) {
	return fma(z, (z * 3.0), (x * y));
}

Julia

function code(x, y, z)
	return fma(z, Float64(z * 3.0), Float64(x * y))
end

Wolfram

code[x_, y_, z_] := N[(z * N[(z * 3.0), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.9%

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

   1. associate-+l+ 97.9%

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

   2. +-commutative 97.9%

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

   3. +-commutative 97.9%

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

   4. associate-+r+ 97.5%

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

   5. distribute-lft-out 97.5%

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

   6. distribute-lft-out 97.5%

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

   7. fma-def 99.1%

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

   8. remove-double-neg 99.1%

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

   9. unsub-neg 99.1%

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

   10. count-2 99.1%

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

   11. neg-mul-1 99.1%

       \[\leadsto \mathsf{fma}\left(z, 2 \cdot z - \color{blue}{-1 \cdot z}, x \cdot y\right) \]

   12. distribute-rgt-out-- 99.1%

       \[\leadsto \mathsf{fma}\left(z, \color{blue}{z \cdot \left(2 - -1\right)}, x \cdot y\right) \]

   13. metadata-eval 99.1%

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

3. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma x y (* 3.0 (* z z))))

C

double code(double x, double y, double z) {
	return fma(x, y, (3.0 * (z * z)));
}

Julia

function code(x, y, z)
	return fma(x, y, Float64(3.0 * Float64(z * z)))
end

Wolfram

code[x_, y_, z_] := N[(x * y + N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.9%

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

   1. associate-+l+ 97.9%

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

   2. associate-+l+ 97.5%

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

   3. fma-def 98.3%

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

   4. cancel-sign-sub 98.3%

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

   5. neg-mul-1 98.3%

      \[\leadsto \mathsf{fma}\left(x, y, \left(z \cdot z + z \cdot z\right) - \color{blue}{\left(-1 \cdot z\right)} \cdot z\right) \]

   6. associate-*l* 98.3%

      \[\leadsto \mathsf{fma}\left(x, y, \left(z \cdot z + z \cdot z\right) - \color{blue}{-1 \cdot \left(z \cdot z\right)}\right) \]

   7. count-2 98.3%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{2 \cdot \left(z \cdot z\right)} - -1 \cdot \left(z \cdot z\right)\right) \]

   8. distribute-rgt-out-- 98.3%

      \[\leadsto \mathsf{fma}\left(x, y, \color{blue}{\left(z \cdot z\right) \cdot \left(2 - -1\right)}\right) \]

   9. metadata-eval 98.3%

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

3. Simplified 98.3%

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

4. Final simplification 98.3%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

2. Final simplification 97.9%

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

Alternative 4: 78.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around inf 76.5%

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

3. Final simplification 76.5%

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

Alternative 5: 78.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around inf 76.5%

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

3. Taylor expanded in x around inf 75.9%

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

4. Final simplification 75.9%

   \[\leadsto x \cdot y + z \cdot z \]

Alternative 6: 53.2% 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 97.9%

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

2. Taylor expanded in x around 0 97.9%

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

4. Simplified 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left({z}^{2}, 3, x \cdot y\right)} \]

5. Taylor expanded in z around 0 53.2%

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

6. Final simplification 53.2%

   \[\leadsto x \cdot y \]

Developer target: 98.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot z\right) \cdot z + y \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (* 3.0 z) z) (* y x)))

C

double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (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 = ((3.0d0 * z) * z) + (y * x)
end function

Java

public static double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * x);
}

Python

def code(x, y, z):
	return ((3.0 * z) * z) + (y * x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(3.0 * z) * z) + Float64(y * x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((3.0 * z) * z) + (y * x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023314 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

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