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

Percentage Accurate: 98.4% → 99.7%

Time: 7.3s

Alternatives: 8

Speedup: 0.7×

• 34.4% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+303}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(x \cdot y + z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{\frac{z \cdot 3}{x}}{\frac{1}{z}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+303)
   (+ (* z z) (+ (* z z) (+ (* x y) (* z z))))
   (* x (+ y (/ (/ (* z 3.0) x) (/ 1.0 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+303) {
		tmp = (z * z) + ((z * z) + ((x * y) + (z * z)));
	} else {
		tmp = x * (y + (((z * 3.0) / x) / (1.0 / z)));
	}
	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+303) then
        tmp = (z * z) + ((z * z) + ((x * y) + (z * z)))
    else
        tmp = x * (y + (((z * 3.0d0) / x) / (1.0d0 / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+303) {
		tmp = (z * z) + ((z * z) + ((x * y) + (z * z)));
	} else {
		tmp = x * (y + (((z * 3.0) / x) / (1.0 / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+303:
		tmp = (z * z) + ((z * z) + ((x * y) + (z * z)))
	else:
		tmp = x * (y + (((z * 3.0) / x) / (1.0 / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+303)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(x * y) + Float64(z * z))));
	else
		tmp = Float64(x * Float64(y + Float64(Float64(Float64(z * 3.0) / x) / Float64(1.0 / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+303)
		tmp = (z * z) + ((z * z) + ((x * y) + (z * z)));
	else
		tmp = x * (y + (((z * 3.0) / x) / (1.0 / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+303], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + N[(N[(N[(z * 3.0), $MachinePrecision] / x), $MachinePrecision] / N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e303

   1. Initial program 99.8%

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

   if 1e303 < (*.f64 z z)

   1. Initial program 90.0%

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

      1. associate-+l+ 90.0%

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

      2. associate-+l+ 90.0%

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

      3. +-commutative 90.0%

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

      4. distribute-lft-out 90.0%

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

      5. distribute-lft-out 90.0%

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

      6. fma-define 95.0%

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

      7. remove-double-neg 95.0%

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

      8. unsub-neg 95.0%

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

      9. count-2 95.0%

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

      10. neg-mul-1 95.0%

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

      11. distribute-rgt-out-- 95.0%

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

      12. metadata-eval 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in x around inf 95.0%

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

      1. unpow2 95.0%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-*r/ 100.0%

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

       2. div-inv 100.0%

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

       3. associate-/r* 100.0%

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

   11. Applied egg-rr 100.0%

       \[\leadsto x \cdot \left(y + \color{blue}{\frac{\frac{3 \cdot z}{x}}{\frac{1}{z}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+303}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(x \cdot y + z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + \frac{\frac{z \cdot 3}{x}}{\frac{1}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 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 96.7%

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

   1. associate-+l+ 96.8%

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

   2. associate-+l+ 96.8%

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

   3. +-commutative 96.8%

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

   4. distribute-lft-out 96.8%

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

   5. distribute-lft-out 96.8%

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

   6. fma-define 98.3%

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

   7. remove-double-neg 98.3%

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

   8. unsub-neg 98.3%

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

   9. count-2 98.3%

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

   10. neg-mul-1 98.3%

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

   11. distribute-rgt-out-- 98.3%

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

   12. metadata-eval 98.3%

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

3. Simplified 98.3%

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

Alternative 3: 86.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10000000000000:\\ \;\;\;\;x \cdot y + z \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 10000000000000.0) (+ (* x y) (* z z)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 10000000000000.0) {
		tmp = (x * y) + (z * z);
	} 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) <= 10000000000000.0d0) then
        tmp = (x * y) + (z * z)
    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) <= 10000000000000.0) {
		tmp = (x * y) + (z * z);
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 10000000000000.0:
		tmp = (x * y) + (z * z)
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 10000000000000.0)
		tmp = Float64(Float64(x * y) + Float64(z * z));
	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) <= 10000000000000.0)
		tmp = (x * y) + (z * z);
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 10000000000000.0], N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e13

   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 89.9%

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

   4. Taylor expanded in x around inf 89.6%

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

   if 1e13 < (*.f64 z z)

   1. Initial program 93.9%

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

   3. Taylor expanded in x around 0 88.3%

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

      1. distribute-lft1-in 88.3%

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

      2. metadata-eval 88.3%

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

      3. *-commutative 88.3%

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

   5. Simplified 88.3%

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

      1. *-commutative 88.3%

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

      2. unpow2 88.3%

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

      3. associate-*r* 88.2%

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

      4. *-commutative 88.2%

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

   7. Applied egg-rr 88.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10000000000000:\\ \;\;\;\;x \cdot y + z \cdot z\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y + \frac{\frac{z \cdot 3}{x}}{\frac{1}{z}}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (+ y (/ (/ (* z 3.0) x) (/ 1.0 z)))))

C

double code(double x, double y, double z) {
	return x * (y + (((z * 3.0) / x) / (1.0 / 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 * 3.0d0) / x) / (1.0d0 / z)))
end function

Java

public static double code(double x, double y, double z) {
	return x * (y + (((z * 3.0) / x) / (1.0 / z)));
}

Python

def code(x, y, z):
	return x * (y + (((z * 3.0) / x) / (1.0 / z)))

Julia

function code(x, y, z)
	return Float64(x * Float64(y + Float64(Float64(Float64(z * 3.0) / x) / Float64(1.0 / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * (y + (((z * 3.0) / x) / (1.0 / z)));
end

Wolfram

code[x_, y_, z_] := N[(x * N[(y + N[(N[(N[(z * 3.0), $MachinePrecision] / x), $MachinePrecision] / N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

   1. associate-+l+ 96.8%

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

   2. associate-+l+ 96.8%

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

   3. +-commutative 96.8%

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

   4. distribute-lft-out 96.8%

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

   5. distribute-lft-out 96.8%

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

   6. fma-define 98.3%

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

   7. remove-double-neg 98.3%

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

   8. unsub-neg 98.3%

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

   9. count-2 98.3%

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

   10. neg-mul-1 98.3%

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

   11. distribute-rgt-out-- 98.3%

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

   12. metadata-eval 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around inf 92.5%

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

   1. unpow2 92.5%

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

   2. associate-/l* 94.1%

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

7. Applied egg-rr 94.1%

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

   1. clear-num 94.0%

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

   2. un-div-inv 94.1%

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

9. Applied egg-rr 94.1%

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

    1. associate-*r/ 94.1%

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

    2. div-inv 94.1%

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

    3. associate-/r* 94.1%

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

11. Applied egg-rr 94.1%

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

12. Final simplification 94.1%

    \[\leadsto x \cdot \left(y + \frac{\frac{z \cdot 3}{x}}{\frac{1}{z}}\right) \]
13. Add Preprocessing

Alternative 5: 93.5% 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 96.7%

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

   1. associate-+l+ 96.8%

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

   2. associate-+l+ 96.8%

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

   3. +-commutative 96.8%

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

   4. distribute-lft-out 96.8%

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

   5. distribute-lft-out 96.8%

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

   6. fma-define 98.3%

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

   7. remove-double-neg 98.3%

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

   8. unsub-neg 98.3%

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

   9. count-2 98.3%

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

   10. neg-mul-1 98.3%

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

   11. distribute-rgt-out-- 98.3%

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

   12. metadata-eval 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around inf 92.5%

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

   1. unpow2 92.5%

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

   2. associate-/l* 94.1%

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

7. Applied egg-rr 94.1%

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

   1. clear-num 94.0%

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

   2. un-div-inv 94.1%

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

9. Applied egg-rr 94.1%

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

Alternative 6: 93.5% 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 96.7%

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

   1. associate-+l+ 96.8%

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

   2. associate-+l+ 96.8%

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

   3. +-commutative 96.8%

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

   4. distribute-lft-out 96.8%

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

   5. distribute-lft-out 96.8%

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

   6. fma-define 98.3%

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

   7. remove-double-neg 98.3%

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

   8. unsub-neg 98.3%

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

   9. count-2 98.3%

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

   10. neg-mul-1 98.3%

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

   11. distribute-rgt-out-- 98.3%

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

   12. metadata-eval 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around inf 92.5%

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

   1. unpow2 92.5%

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

   2. associate-/l* 94.1%

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

7. Applied egg-rr 94.1%

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

Alternative 7: 69.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.42 \cdot 10^{-22}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.42e-22) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.42e-22) {
		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 <= 1.42d-22) 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 <= 1.42e-22) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.42e-22:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.42e-22)
		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 <= 1.42e-22)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.42e-22], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4200000000000001e-22

   1. Initial program 97.2%

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

      1. associate-+l+ 97.2%

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

      2. associate-+l+ 97.2%

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

      3. +-commutative 97.2%

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

      4. distribute-lft-out 97.2%

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

      5. distribute-lft-out 97.2%

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

      6. fma-define 97.7%

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

      7. remove-double-neg 97.7%

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

      8. unsub-neg 97.7%

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

      9. count-2 97.7%

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

      10. neg-mul-1 97.7%

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

      11. distribute-rgt-out-- 97.7%

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

      12. metadata-eval 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in x around inf 95.9%

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

   6. Taylor expanded in x around inf 60.3%

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

   if 1.4200000000000001e-22 < z

   1. Initial program 95.7%

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

   3. Taylor expanded in x around 0 77.6%

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

      1. distribute-lft1-in 77.6%

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

      2. metadata-eval 77.6%

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

      3. *-commutative 77.6%

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

   5. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. unpow2 77.6%

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

      3. associate-*r* 77.5%

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

      4. *-commutative 77.5%

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

   7. Applied egg-rr 77.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.42 \cdot 10^{-22}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 96.7%

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

   1. associate-+l+ 96.8%

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

   2. associate-+l+ 96.8%

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

   3. +-commutative 96.8%

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

   4. distribute-lft-out 96.8%

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

   5. distribute-lft-out 96.8%

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

   6. fma-define 98.3%

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

   7. remove-double-neg 98.3%

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

   8. unsub-neg 98.3%

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

   9. count-2 98.3%

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

   10. neg-mul-1 98.3%

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

   11. distribute-rgt-out-- 98.3%

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

   12. metadata-eval 98.3%

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

3. Simplified 98.3%

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

5. Taylor expanded in x around inf 92.5%

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

6. Taylor expanded in x around inf 50.3%

   \[\leadsto \color{blue}{x \cdot y} \]
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 2024096 
(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)))