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

Percentage Accurate: 98.4% → 99.3%

Time: 7.4s

Alternatives: 7

Speedup: 1.0×

• 34.3% 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: 99.3% 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-def 99.1%

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

   3. associate-+l+ 99.1%

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

   4. fma-def 99.9%

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

   5. count-2 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.3% 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-def 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. Final simplification 99.9%

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

Alternative 3: 86.1% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+21) (+ (* z z) (+ (* z z) (* x y))) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+21) {
		tmp = (z * z) + ((z * z) + (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 * z) <= 1d+21) then
        tmp = (z * z) + ((z * z) + (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 * z) <= 1e+21) {
		tmp = (z * z) + ((z * z) + (x * y));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+21:
		tmp = (z * z) + ((z * z) + (x * y))
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+21)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + 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 * z) <= 1e+21)
		tmp = (z * z) + ((z * z) + (x * y));
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+21], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e21

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 86.3%

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

   if 1e21 < (*.f64 z z)

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 88.4%

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

   5. Simplified 88.4%

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

      1. add-sqr-sqrt 88.3%

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

      2. pow2 88.3%

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

      3. sqrt-prod 88.2%

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

      4. unpow2 88.2%

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

      5. sqrt-prod 47.4%

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

      6. add-sqr-sqrt 88.2%

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

   7. Applied egg-rr 88.2%

      \[\leadsto \color{blue}{{\left(z \cdot \sqrt{3}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 88.2%

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

      2. *-commutative 88.2%

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

      3. *-commutative 88.2%

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

      4. swap-sqr 88.1%

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

      5. metadata-eval 88.1%

         \[\leadsto \left(\sqrt{\color{blue}{2 + 1}} \cdot \sqrt{3}\right) \cdot \left(z \cdot z\right) \]

      6. metadata-eval 88.1%

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

      7. add-sqr-sqrt 88.4%

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

      8. metadata-eval 88.4%

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

      9. metadata-eval 88.4%

         \[\leadsto \left({2}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right) \cdot \left(z \cdot z\right) \]

      10. sqrt-pow2 88.0%

          \[\leadsto \left(\color{blue}{{\left(\sqrt{2}\right)}^{2}} + 1\right) \cdot \left(z \cdot z\right) \]

      11. associate-*r* 88.0%

          \[\leadsto \color{blue}{\left(\left({\left(\sqrt{2}\right)}^{2} + 1\right) \cdot z\right) \cdot z} \]

      12. sqrt-pow2 88.4%

          \[\leadsto \left(\left(\color{blue}{{2}^{\left(\frac{2}{2}\right)}} + 1\right) \cdot z\right) \cdot z \]

      13. metadata-eval 88.4%

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

      14. metadata-eval 88.4%

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

      15. metadata-eval 88.4%

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

   9. Applied egg-rr 88.4%

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

4. Final simplification 87.2%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $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. Final simplification 99.1%

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

Alternative 5: 85.9% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+21) (+ (* z z) (* x y)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+21) {
		tmp = (z * z) + (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 * z) <= 1d+21) then
        tmp = (z * z) + (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 * z) <= 1e+21) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+21:
		tmp = (z * z) + (x * y)
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+21)
		tmp = Float64(Float64(z * z) + 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 * z) <= 1e+21)
		tmp = (z * z) + (x * y);
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+21], N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e21

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 86.3%

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

   4. Taylor expanded in x around inf 85.9%

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

   if 1e21 < (*.f64 z z)

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0 88.4%

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

   5. Simplified 88.4%

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

      1. add-sqr-sqrt 88.3%

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

      2. pow2 88.3%

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

      3. sqrt-prod 88.2%

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

      4. unpow2 88.2%

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

      5. sqrt-prod 47.4%

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

      6. add-sqr-sqrt 88.2%

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

   7. Applied egg-rr 88.2%

      \[\leadsto \color{blue}{{\left(z \cdot \sqrt{3}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 88.2%

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

      2. *-commutative 88.2%

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

      3. *-commutative 88.2%

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

      4. swap-sqr 88.1%

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

      5. metadata-eval 88.1%

         \[\leadsto \left(\sqrt{\color{blue}{2 + 1}} \cdot \sqrt{3}\right) \cdot \left(z \cdot z\right) \]

      6. metadata-eval 88.1%

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

      7. add-sqr-sqrt 88.4%

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

      8. metadata-eval 88.4%

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

      9. metadata-eval 88.4%

         \[\leadsto \left({2}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right) \cdot \left(z \cdot z\right) \]

      10. sqrt-pow2 88.0%

          \[\leadsto \left(\color{blue}{{\left(\sqrt{2}\right)}^{2}} + 1\right) \cdot \left(z \cdot z\right) \]

      11. associate-*r* 88.0%

          \[\leadsto \color{blue}{\left(\left({\left(\sqrt{2}\right)}^{2} + 1\right) \cdot z\right) \cdot z} \]

      12. sqrt-pow2 88.4%

          \[\leadsto \left(\left(\color{blue}{{2}^{\left(\frac{2}{2}\right)}} + 1\right) \cdot z\right) \cdot z \]

      13. metadata-eval 88.4%

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

      14. metadata-eval 88.4%

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

      15. metadata-eval 88.4%

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

   9. Applied egg-rr 88.4%

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

4. Final simplification 87.0%

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

Alternative 6: 69.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 34000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 34000000000.0) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 34000000000.0) {
		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 <= 34000000000.0d0) 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 <= 34000000000.0) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 34000000000.0:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 34000000000.0)
		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 <= 34000000000.0)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 34000000000.0], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.4e10

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-def 99.9%

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

      3. associate-+l+ 99.9%

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

      4. fma-def 99.9%

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

      5. count-2 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. *-commutative 99.9%

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

      4. sqrt-prod 99.7%

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

      5. sqrt-prod 36.5%

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

      6. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around 0 65.6%

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

   if 3.4e10 < z

   1. Initial program 96.6%

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

   3. Taylor expanded in x around 0 84.3%

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

   5. Simplified 84.3%

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

      1. add-sqr-sqrt 84.3%

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

      2. pow2 84.3%

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

      3. sqrt-prod 84.3%

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

      4. unpow2 84.3%

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

      5. sqrt-prod 84.0%

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

      6. add-sqr-sqrt 84.3%

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

   7. Applied egg-rr 84.3%

      \[\leadsto \color{blue}{{\left(z \cdot \sqrt{3}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 84.3%

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

      2. *-commutative 84.3%

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

      3. *-commutative 84.3%

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

      4. swap-sqr 84.1%

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

      5. metadata-eval 84.1%

         \[\leadsto \left(\sqrt{\color{blue}{2 + 1}} \cdot \sqrt{3}\right) \cdot \left(z \cdot z\right) \]

      6. metadata-eval 84.1%

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

      7. add-sqr-sqrt 84.3%

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

      8. metadata-eval 84.3%

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

      9. metadata-eval 84.3%

         \[\leadsto \left({2}^{\color{blue}{\left(\frac{2}{2}\right)}} + 1\right) \cdot \left(z \cdot z\right) \]

      10. sqrt-pow2 84.0%

          \[\leadsto \left(\color{blue}{{\left(\sqrt{2}\right)}^{2}} + 1\right) \cdot \left(z \cdot z\right) \]

      11. associate-*r* 84.1%

          \[\leadsto \color{blue}{\left(\left({\left(\sqrt{2}\right)}^{2} + 1\right) \cdot z\right) \cdot z} \]

      12. sqrt-pow2 84.4%

          \[\leadsto \left(\left(\color{blue}{{2}^{\left(\frac{2}{2}\right)}} + 1\right) \cdot z\right) \cdot z \]

      13. metadata-eval 84.4%

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

      14. metadata-eval 84.4%

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

      15. metadata-eval 84.4%

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

   9. Applied egg-rr 84.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 34000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.3% 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. 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-def 99.1%

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

   3. associate-+l+ 99.1%

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

   4. fma-def 99.9%

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

   5. count-2 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 99.8%

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

   2. pow2 99.8%

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

   3. *-commutative 99.8%

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

   4. sqrt-prod 99.7%

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

   5. sqrt-prod 51.8%

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

   6. add-sqr-sqrt 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in z around 0 54.0%

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

8. Final simplification 54.0%

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

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 2024019 
(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)))