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

Percentage Accurate: 98.3% → 99.3%

Time: 9.3s

Alternatives: 8

Speedup: 1.0×

• 34.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * y) + (z * z)) + (z * z)) + (z * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% 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: 99.3% 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-define 99.2%

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

   3. associate-+l+ 99.2%

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

   4. fma-define 99.6%

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

   5. count-2 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 99.3% 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-define 99.5%

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

   4. distribute-lft-out 99.5%

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

   5. distribute-lft-out 99.5%

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

   6. count-2 99.5%

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

   7. distribute-rgt1-in 99.5%

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

   8. metadata-eval 99.5%

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

   9. *-commutative 99.5%

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

   10. metadata-eval 99.5%

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

   11. metadata-eval 99.5%

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

3. Simplified 99.5%

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

Alternative 3: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-205}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \frac{z \cdot z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{z \cdot 3}{\frac{y}{z}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.6e-205)
   (* x (+ y (* 3.0 (/ (* z z) x))))
   (* y (+ x (/ (* z 3.0) (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.6e-205) {
		tmp = x * (y + (3.0 * ((z * z) / x)));
	} else {
		tmp = y * (x + ((z * 3.0) / (y / 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 (y <= 4.6d-205) then
        tmp = x * (y + (3.0d0 * ((z * z) / x)))
    else
        tmp = y * (x + ((z * 3.0d0) / (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.6e-205) {
		tmp = x * (y + (3.0 * ((z * z) / x)));
	} else {
		tmp = y * (x + ((z * 3.0) / (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.6e-205:
		tmp = x * (y + (3.0 * ((z * z) / x)))
	else:
		tmp = y * (x + ((z * 3.0) / (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.6e-205)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(Float64(z * z) / x))));
	else
		tmp = Float64(y * Float64(x + Float64(Float64(z * 3.0) / Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.6e-205)
		tmp = x * (y + (3.0 * ((z * z) / x)));
	else
		tmp = y * (x + ((z * 3.0) / (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.6e-205], N[(x * N[(y + N[(3.0 * N[(N[(z * z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(N[(z * 3.0), $MachinePrecision] / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5999999999999998e-205

   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 93.9%

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

   5. Simplified 93.9%

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

      1. unpow2 93.9%

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

   7. Applied egg-rr 93.9%

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

   if 4.5999999999999998e-205 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in y around inf 98.8%

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

   5. Simplified 98.8%

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

      1. div-inv 98.8%

         \[\leadsto y \cdot \left(x + \color{blue}{\left({z}^{2} \cdot \frac{1}{y}\right)} \cdot 3\right) \]

      2. unpow2 98.8%

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

      3. associate-*l* 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto y \cdot \left(x + \color{blue}{\left(z \cdot \left(z \cdot \frac{1}{y}\right)\right)} \cdot 3\right) \]
   8. Step-by-step derivation

      1. un-div-inv 99.9%

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

      2. clear-num 99.9%

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

   9. Applied egg-rr 99.9%

      \[\leadsto y \cdot \left(x + \left(z \cdot \color{blue}{\frac{1}{\frac{y}{z}}}\right) \cdot 3\right) \]
   10. Step-by-step derivation

       1. un-div-inv 99.9%

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

       2. associate-*l/ 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-205}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \frac{z \cdot z}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{z \cdot 3}{\frac{y}{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-120}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 3 \cdot \frac{z \cdot z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-120)
   (* 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 (x <= -1.5e-120) {
		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 (x <= (-1.5d-120)) 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 (x <= -1.5e-120) {
		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 x <= -1.5e-120:
		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 (x <= -1.5e-120)
		tmp = Float64(x * Float64(y + Float64(3.0 * Float64(z * Float64(z / x)))));
	else
		tmp = Float64(y * Float64(x + Float64(3.0 * Float64(Float64(z * z) / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5e-120)
		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[x, -1.5e-120], 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[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000005e-120

   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 96.5%

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

   5. Simplified 96.5%

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

      1. unpow2 96.5%

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

      2. associate-/l* 97.7%

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

   7. Applied egg-rr 97.7%

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

   if -1.50000000000000005e-120 < x

   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 y around inf 97.6%

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

   5. Simplified 97.6%

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

      1. unpow2 93.3%

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

   7. Applied egg-rr 97.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-120}:\\ \;\;\;\;x \cdot \left(y + 3 \cdot \left(z \cdot \frac{z}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + 3 \cdot \frac{z \cdot z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% 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 6: 82.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3.25 \cdot 10^{-142}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 3.25e-142) (* x y) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 3.25e-142) {
		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) <= 3.25d-142) 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) <= 3.25e-142) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 3.25e-142:
		tmp = x * y
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 3.25e-142)
		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) <= 3.25e-142)
		tmp = x * y;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 3.25e-142], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.25000000000000013e-142

   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 91.2%

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

   if 3.25000000000000013e-142 < (*.f64 z z)

   1. Initial program 98.6%

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

   3. Taylor expanded in x around 0 75.4%

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

   5. Simplified 75.4%

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

      1. unpow2 91.0%

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

   7. Applied egg-rr 75.4%

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

Alternative 7: 94.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * (y + (3.0d0 * (z * (z / x))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in x around inf 94.4%

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

5. Simplified 94.4%

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

   1. unpow2 94.4%

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

   2. associate-/l* 94.7%

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

7. Applied egg-rr 94.7%

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

8. Final simplification 94.7%

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

Alternative 8: 52.3% 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. Add Preprocessing

3. Taylor expanded in x around inf 51.7%

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

Developer Target 1: 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 2024177 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* 3 z) z) (* y x)))

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