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

Percentage Accurate: 98.3% → 99.1%

Time: 8.3s

Alternatives: 8

Speedup: 0.7×

• 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: 98.3% 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: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+285}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+285)
   (+ (* z z) (+ (* z z) (+ (* z z) (* x y))))
   (* x (+ y (* 3.0 (* z (/ z x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+285) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = x * (y + (3.0 * (z * (z / x))));
	}
	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) <= 2d+285) then
        tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
    else
        tmp = x * (y + (3.0d0 * (z * (z / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+285) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = x * (y + (3.0 * (z * (z / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+285:
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
	else:
		tmp = x * (y + (3.0 * (z * (z / x))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+285)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y))));
	else
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z * Float64(z / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+285)
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	else
		tmp = x * (y + (3.0 * (z * (z / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+285], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y + N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2e285

   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 2e285 < (*.f64 z z)

   1. Initial program 91.0%

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

   3. Taylor expanded in x around inf 97.0%

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

      1. distribute-lft1-in 97.0%

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

      2. metadata-eval 97.0%

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

   5. Simplified 97.0%

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

      1. unpow2 97.0%

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

      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%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+285}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% 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 97.5%

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

   1. +-commutative 97.5%

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

   2. fma-define 97.6%

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

   3. associate-+l+ 97.6%

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

   4. fma-define 99.2%

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

   5. count-2 99.2%

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

3. Simplified 99.2%

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

Alternative 3: 99.4% 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 97.5%

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

   1. associate-+l+ 97.5%

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

   2. associate-+l+ 97.5%

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

   3. fma-define 99.1%

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

   4. associate-+r+ 99.1%

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

   5. distribute-lft-out 99.1%

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

   6. distribute-lft-out 99.1%

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

   7. remove-double-neg 99.1%

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

   8. unsub-neg 99.1%

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

   9. count-2 99.1%

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

   10. neg-mul-1 99.1%

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

   11. distribute-rgt-out-- 99.1%

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

   12. metadata-eval 99.1%

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

3. Simplified 99.1%

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

Alternative 4: 83.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.4 \cdot 10^{-134} \lor \neg \left(z \cdot z \leq 1.6 \cdot 10^{-91}\right) \land z \cdot z \leq 3.1 \cdot 10^{-41}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 1.4e-134)
         (and (not (<= (* z z) 1.6e-91)) (<= (* z z) 3.1e-41)))
   (* x y)
   (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 1.4e-134) || (!((z * z) <= 1.6e-91) && ((z * z) <= 3.1e-41))) {
		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 * z) <= 1.4d-134) .or. (.not. ((z * z) <= 1.6d-91)) .and. ((z * z) <= 3.1d-41)) 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 * z) <= 1.4e-134) || (!((z * z) <= 1.6e-91) && ((z * z) <= 3.1e-41))) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 1.4e-134) or (not ((z * z) <= 1.6e-91) and ((z * z) <= 3.1e-41)):
		tmp = x * y
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(z * z) <= 1.4e-134) || (!(Float64(z * z) <= 1.6e-91) && (Float64(z * z) <= 3.1e-41)))
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((z * z) <= 1.4e-134) || (~(((z * z) <= 1.6e-91)) && ((z * z) <= 3.1e-41)))
		tmp = x * y;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 1.4e-134], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 1.6e-91]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 3.1e-41]]], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.3999999999999999e-134 or 1.59999999999999998e-91 < (*.f64 z z) < 3.10000000000000001e-41

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. distribute-lft1-in 100.0%

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

      2. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 91.8%

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

   if 1.3999999999999999e-134 < (*.f64 z z) < 1.59999999999999998e-91 or 3.10000000000000001e-41 < (*.f64 z z)

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0 85.3%

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

      1. distribute-lft1-in 85.3%

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

      2. metadata-eval 85.3%

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

      3. *-commutative 85.3%

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

   5. Simplified 85.3%

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

      1. unpow2 85.3%

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

   7. Applied egg-rr 85.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.4 \cdot 10^{-134} \lor \neg \left(z \cdot z \leq 1.6 \cdot 10^{-91}\right) \land z \cdot z \leq 3.1 \cdot 10^{-41}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+98) (* x (+ y (* 3.0 (* z (/ z x))))) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4e+98) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} 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) <= 4d+98) then
        tmp = x * (y + (3.0d0 * (z * (z / x))))
    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) <= 4e+98) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e+98:
		tmp = x * (y + (3.0 * (z * (z / x))))
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4e+98)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z * Float64(z / x)))));
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 4e+98)
		tmp = x * (y + (3.0 * (z * (z / x))));
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+98], N[(x * N[(y + N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.99999999999999999e98

   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 98.6%

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

      1. distribute-lft1-in 98.6%

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

      2. metadata-eval 98.6%

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

   5. Simplified 98.6%

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

      1. unpow2 98.6%

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

      2. associate-/l* 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 3.99999999999999999e98 < (*.f64 z z)

   1. Initial program 94.4%

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

   3. Taylor expanded in x around 0 91.7%

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

      1. distribute-lft1-in 91.7%

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

      2. metadata-eval 91.7%

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

      3. *-commutative 91.7%

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

   5. Simplified 91.7%

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

      1. unpow2 91.7%

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

   7. Applied egg-rr 91.7%

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

Alternative 6: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-138}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 3 \cdot \left(z \cdot \frac{z}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-138)
   (* x (+ y (* 3.0 (* z (/ z x)))))
   (* y (+ x (* 3.0 (* z (/ z y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-138) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + (3.0 * (z * (z / y))));
	}
	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 (y <= 2d-138) then
        tmp = x * (y + (3.0d0 * (z * (z / x))))
    else
        tmp = y * (x + (3.0d0 * (z * (z / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-138) {
		tmp = x * (y + (3.0 * (z * (z / x))));
	} else {
		tmp = y * (x + (3.0 * (z * (z / y))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-138:
		tmp = x * (y + (3.0 * (z * (z / x))))
	else:
		tmp = y * (x + (3.0 * (z * (z / y))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-138)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z * Float64(z / x)))));
	else
		tmp = Float64(y * Float64(x + Float64(3.0 * Float64(z * Float64(z / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-138)
		tmp = x * (y + (3.0 * (z * (z / x))));
	else
		tmp = y * (x + (3.0 * (z * (z / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-138], N[(x * N[(y + N[(3.0 * N[(z * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(3.0 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000013e-138

   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 inf 91.8%

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

      1. distribute-lft1-in 91.8%

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

      2. metadata-eval 91.8%

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

   5. Simplified 91.8%

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

      1. unpow2 91.8%

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

      2. associate-/l* 93.1%

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

   7. Applied egg-rr 93.1%

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

   if 2.00000000000000013e-138 < y

   1. Initial program 98.9%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. distribute-lft1-in 99.8%

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

      2. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. associate-/l* 99.8%

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

   7. Applied egg-rr 99.8%

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

Alternative 7: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 6.5 \cdot 10^{-39}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 6.5e-39) (+ (* z z) (* x y)) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 6.5e-39) {
		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) <= 6.5d-39) 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) <= 6.5e-39) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 6.5e-39:
		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) <= 6.5e-39)
		tmp = Float64(Float64(z * z) + Float64(x * y));
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 6.5e-39)
		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], 6.5e-39], N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 6.50000000000000027e-39

   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. Step-by-step derivation

      1. associate-+l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. count-2 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. fma-define 99.9%

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

      6. *-commutative 99.9%

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

      7. sqrt-prod 99.9%

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

      8. sqrt-prod 48.6%

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

      9. add-sqr-sqrt 92.9%

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

      10. *-commutative 92.9%

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

      11. sqrt-prod 92.8%

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

      12. sqrt-prod 48.6%

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

      13. add-sqr-sqrt 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-/l* 99.8%

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

      3. fma-define 99.8%

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

      4. unpow2 99.8%

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

      5. rem-square-sqrt 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in z around 0 87.7%

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

   if 6.50000000000000027e-39 < (*.f64 z z)

   1. Initial program 95.4%

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

   3. Taylor expanded in x around 0 86.3%

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

      1. distribute-lft1-in 86.3%

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

      2. metadata-eval 86.3%

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

      3. *-commutative 86.3%

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

   5. Simplified 86.3%

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

      1. unpow2 86.3%

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

   7. Applied egg-rr 86.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 6.5 \cdot 10^{-39}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 97.5%

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

3. Taylor expanded in x around inf 91.3%

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

   1. distribute-lft1-in 91.3%

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

   2. metadata-eval 91.3%

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

5. Simplified 91.3%

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

6. Taylor expanded in x around inf 51.0%

   \[\leadsto \color{blue}{x \cdot y} \]
7. Add Preprocessing

Developer target: 98.3% 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 2024111 
(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)))