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

Percentage Accurate: 98.4% → 99.6%

Time: 8.6s

Alternatives: 10

Speedup: 1.2×

• 33.6% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z z (fma x y (* 2.0 (* z z)))))

C

double code(double x, double y, double z) {
	return fma(z, z, fma(x, y, (2.0 * (z * z))));
}

Julia

function code(x, y, z)
	return fma(z, z, fma(x, y, Float64(2.0 * Float64(z * z))))
end

Wolfram

code[x_, y_, z_] := N[(z * z + N[(x * y + N[(2.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

   1. +-commutative 99.1%

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

   2. fma-define 99.2%

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

   3. associate-+l+ 99.2%

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

   4. fma-define 100.0%

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

   5. count-2 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. associate-+l+ 99.1%

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

   2. associate-+l+ 99.1%

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

   3. fma-define 99.9%

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

   4. associate-+r+ 99.9%

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

   5. distribute-lft-out 99.9%

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

   6. distribute-lft-out 99.9%

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

   7. remove-double-neg 99.9%

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

   8. unsub-neg 99.9%

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

   9. count-2 99.9%

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

   10. neg-mul-1 99.9%

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

   11. distribute-rgt-out-- 99.9%

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

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

Alternative 3: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ z \cdot z + \left(z \cdot \left(z \cdot 2\right) + x \cdot y\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* z z) (+ (* z (* z 2.0)) (* x y))))

C

double code(double x, double y, double z) {
	return (z * z) + ((z * (z * 2.0)) + (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 = (z * z) + ((z * (z * 2.0d0)) + (x * y))
end function

Java

public static double code(double x, double y, double z) {
	return (z * z) + ((z * (z * 2.0)) + (x * y));
}

Python

def code(x, y, z):
	return (z * z) + ((z * (z * 2.0)) + (x * y))

Julia

function code(x, y, z)
	return Float64(Float64(z * z) + Float64(Float64(z * Float64(z * 2.0)) + Float64(x * y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (z * z) + ((z * (z * 2.0)) + (x * y));
end

Wolfram

code[x_, y_, z_] := N[(N[(z * z), $MachinePrecision] + N[(N[(z * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.1%

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

   1. associate-+l+ 99.1%

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

   2. +-commutative 99.1%

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

   3. count-2 99.1%

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

   4. associate-*r* 99.1%

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

   5. add-sqr-sqrt 46.0%

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

   6. associate-*r* 46.0%

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

   7. fma-define 46.0%

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

   8. *-commutative 46.0%

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

4. Applied egg-rr 46.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot 2\right) \cdot \sqrt{z}, \sqrt{z}, x \cdot y\right)} + z \cdot z \]
5. Step-by-step derivation

   1. fma-undefine 46.0%

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

   2. associate-*l* 46.0%

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

   3. associate-*l* 46.0%

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

   4. associate-*r* 46.0%

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

   5. add-sqr-sqrt 99.1%

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

   6. *-commutative 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 4: 93.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in y around inf 92.6%

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

5. Simplified 92.6%

   \[\leadsto \color{blue}{y \cdot \left(x + 3 \cdot \frac{{z}^{2}}{y}\right)} \]
6. Step-by-step derivation

   1. pow2 92.6%

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

   2. associate-/l* 92.6%

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

7. Applied egg-rr 92.6%

   \[\leadsto y \cdot \left(x + 3 \cdot \color{blue}{\left(z \cdot \frac{z}{y}\right)}\right) \]
8. Step-by-step derivation

   1. associate-*r* 92.6%

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

   2. *-commutative 92.6%

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

   3. clear-num 92.6%

      \[\leadsto y \cdot \left(x + \left(z \cdot 3\right) \cdot \color{blue}{\frac{1}{\frac{y}{z}}}\right) \]

   4. un-div-inv 92.6%

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

9. Applied egg-rr 92.6%

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

Alternative 5: 93.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in y around inf 92.6%

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

5. Simplified 92.6%

   \[\leadsto \color{blue}{y \cdot \left(x + 3 \cdot \frac{{z}^{2}}{y}\right)} \]
6. Step-by-step derivation

   1. pow2 92.6%

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

   2. associate-/l* 92.6%

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

7. Applied egg-rr 92.6%

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

Alternative 6: 93.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in x around inf 92.7%

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

5. Simplified 92.7%

   \[\leadsto \color{blue}{x \cdot \left(y + 3 \cdot \frac{{z}^{2}}{x}\right)} \]
6. Step-by-step derivation

   1. pow2 92.7%

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

   2. associate-/l* 92.7%

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

7. Applied egg-rr 92.7%

   \[\leadsto x \cdot \left(y + 3 \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)}\right) \]
8. Step-by-step derivation

   1. associate-*r* 92.6%

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

   2. associate-*r/ 92.7%

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

   3. *-commutative 92.7%

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

9. Applied egg-rr 92.7%

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

10. Final simplification 92.7%

    \[\leadsto x \cdot \left(y + \frac{z \cdot \left(z \cdot 3\right)}{x}\right) \]
11. Add Preprocessing

Alternative 7: 93.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in x around inf 92.7%

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

5. Simplified 92.7%

   \[\leadsto \color{blue}{x \cdot \left(y + 3 \cdot \frac{{z}^{2}}{x}\right)} \]
6. Step-by-step derivation

   1. pow2 92.7%

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

   2. associate-/l* 92.7%

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

7. Applied egg-rr 92.7%

   \[\leadsto x \cdot \left(y + 3 \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)}\right) \]
8. Step-by-step derivation

   1. clear-num 92.7%

      \[\leadsto x \cdot \left(y + 3 \cdot \left(z \cdot \color{blue}{\frac{1}{\frac{x}{z}}}\right)\right) \]

   2. un-div-inv 92.7%

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

9. Applied egg-rr 92.7%

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

Alternative 8: 93.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in x around inf 92.7%

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

5. Simplified 92.7%

   \[\leadsto \color{blue}{x \cdot \left(y + 3 \cdot \frac{{z}^{2}}{x}\right)} \]
6. Step-by-step derivation

   1. pow2 92.7%

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

   2. associate-/l* 92.7%

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

7. Applied egg-rr 92.7%

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

Alternative 9: 76.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

   1. associate-+l+ 99.1%

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

   2. +-commutative 99.1%

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

   3. count-2 99.1%

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

   4. associate-*r* 99.1%

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

   5. add-sqr-sqrt 46.0%

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

   6. associate-*r* 46.0%

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

   7. fma-define 46.0%

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

   8. *-commutative 46.0%

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

4. Applied egg-rr 46.0%

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

5. Taylor expanded in z around 0 79.1%

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

6. Final simplification 79.1%

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

Alternative 10: 52.6% 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 99.1%

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

3. Taylor expanded in x around inf 92.7%

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

5. Simplified 92.7%

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

6. Taylor expanded in y around inf 54.1%

   \[\leadsto x \cdot \color{blue}{y} \]
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 2024108 
(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)))