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

Percentage Accurate: 98.4% → 98.4%

Time: 4.1s

Alternatives: 3

Speedup: 1.7×

• 34.0% 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: 98.4% 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: 98.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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 * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. associate-+r+ 98.7%

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

   4. distribute-lft-out 98.7%

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

   5. distribute-lft-out 98.8%

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

   6. remove-double-neg 98.8%

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

   7. unsub-neg 98.8%

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

   8. neg-mul-1 98.8%

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

   9. count-2 98.8%

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

   10. distribute-rgt-out-- 98.8%

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

   11. metadata-eval 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 2: 69.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.26 \cdot 10^{+21}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.26e+21) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.26e+21) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	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) :: tmp
    if (z <= 2.26d+21) then
        tmp = x * y
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.26e+21) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.26e+21:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.26e+21)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.26e+21)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.26e+21], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.26e21

   1. Initial program 99.4%

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

      1. associate-+l+ 99.4%

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

      2. associate-+l+ 99.4%

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

      3. associate-+r+ 99.4%

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

      4. distribute-lft-out 99.4%

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

      5. distribute-lft-out 99.4%

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

      6. remove-double-neg 99.4%

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

      7. unsub-neg 99.4%

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

      8. neg-mul-1 99.4%

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

      9. count-2 99.4%

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

      10. distribute-rgt-out-- 99.4%

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

      11. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 68.7%

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

   if 2.26e21 < z

   1. Initial program 97.0%

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

      1. associate-+l+ 97.0%

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

      2. associate-+l+ 97.0%

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

      3. associate-+r+ 97.0%

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

      4. distribute-lft-out 97.0%

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

      5. distribute-lft-out 97.0%

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

      6. remove-double-neg 97.0%

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

      7. unsub-neg 97.0%

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

      8. neg-mul-1 97.0%

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

      9. count-2 97.0%

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

      10. distribute-rgt-out-- 97.0%

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

      11. metadata-eval 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in x around 0 90.9%

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. metadata-eval 90.9%

         \[\leadsto 3 \cdot {z}^{\color{blue}{\left(\frac{4}{2}\right)}} \]

      2. sqrt-pow1 70.2%

         \[\leadsto 3 \cdot \color{blue}{\sqrt{{z}^{4}}} \]

      3. metadata-eval 70.2%

         \[\leadsto \color{blue}{\sqrt{9}} \cdot \sqrt{{z}^{4}} \]

      4. sqrt-prod 70.2%

         \[\leadsto \color{blue}{\sqrt{9 \cdot {z}^{4}}} \]

      5. *-commutative 70.2%

         \[\leadsto \sqrt{\color{blue}{{z}^{4} \cdot 9}} \]

   7. Applied egg-rr 70.2%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. *-commutative 70.2%

         \[\leadsto \sqrt{\color{blue}{9 \cdot {z}^{4}}} \]

      2. sqrt-prod 70.2%

         \[\leadsto \color{blue}{\sqrt{9} \cdot \sqrt{{z}^{4}}} \]

      3. metadata-eval 70.2%

         \[\leadsto \color{blue}{3} \cdot \sqrt{{z}^{4}} \]

      4. sqrt-pow1 90.9%

         \[\leadsto 3 \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}} \]

      5. metadata-eval 90.9%

         \[\leadsto 3 \cdot {z}^{\color{blue}{2}} \]

      6. unpow2 90.9%

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

      7. associate-*r* 90.9%

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

   9. Applied egg-rr 90.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.26 \cdot 10^{+21}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.0% 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 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. associate-+r+ 98.7%

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

   4. distribute-lft-out 98.7%

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

   5. distribute-lft-out 98.8%

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

   6. remove-double-neg 98.8%

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

   7. unsub-neg 98.8%

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

   8. neg-mul-1 98.8%

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

   9. count-2 98.8%

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

   10. distribute-rgt-out-- 98.8%

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

   11. metadata-eval 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in x around inf 54.0%

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

6. Final simplification 54.0%

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

Developer target: 98.4% 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 2024048 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

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

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