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

Percentage Accurate: 98.3% → 99.8%

Time: 7.1s

Alternatives: 6

Speedup: 0.7×

• 34.8% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.35 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(x, y, z \cdot \left(z \cdot 3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{1}{\frac{\frac{y}{z}}{z \cdot 3}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2.35e-23)
   (fma x y (* z (* z 3.0)))
   (* y (+ x (/ 1.0 (/ (/ y z) (* z 3.0)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.35e-23) {
		tmp = fma(x, y, (z * (z * 3.0)));
	} else {
		tmp = y * (x + (1.0 / ((y / z) / (z * 3.0))));
	}
	return tmp;
}

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2.35e-23)
		tmp = fma(x, y, Float64(z * Float64(z * 3.0)));
	else
		tmp = Float64(y * Float64(x + Float64(1.0 / Float64(Float64(y / z) / Float64(z * 3.0)))));
	end
	return tmp
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2.35e-23], N[(x * y + N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(1.0 / N[(N[(y / z), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.35e-23

   1. Initial program 97.8%

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

      1. associate-+l+ 97.8%

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

      2. associate-+l+ 97.8%

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

      3. fma-define 98.8%

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

      4. associate-+r+ 98.8%

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

      5. distribute-lft-out 98.8%

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

      6. distribute-lft-out 98.9%

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

      7. remove-double-neg 98.9%

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

      8. unsub-neg 98.9%

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

      9. count-2 98.9%

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

      10. neg-mul-1 98.9%

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

      11. distribute-rgt-out-- 98.9%

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

      12. metadata-eval 98.9%

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

   3. Simplified 98.9%

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

   if 2.35e-23 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf 96.7%

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

   5. Simplified 96.7%

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

      1. unpow2 96.7%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. associate-*l/ 99.9%

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

       2. clear-num 100.0%

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

   11. Applied egg-rr 100.0%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(y \cdot x + z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{1}{\frac{\frac{y}{z}}{z \cdot 3}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-21)
   (+ (* z z) (+ (* z z) (+ (* y x) (* z z))))
   (* y (+ x (/ 1.0 (/ (/ y z) (* z 3.0)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-21) {
		tmp = (z * z) + ((z * z) + ((y * x) + (z * z)));
	} else {
		tmp = y * (x + (1.0 / ((y / z) / (z * 3.0))));
	}
	return tmp;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
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-21) then
        tmp = (z * z) + ((z * z) + ((y * x) + (z * z)))
    else
        tmp = y * (x + (1.0d0 / ((y / z) / (z * 3.0d0))))
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-21) {
		tmp = (z * z) + ((z * z) + ((y * x) + (z * z)));
	} else {
		tmp = y * (x + (1.0 / ((y / z) / (z * 3.0))));
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if y <= 2e-21:
		tmp = (z * z) + ((z * z) + ((y * x) + (z * z)))
	else:
		tmp = y * (x + (1.0 / ((y / z) / (z * 3.0))))
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-21)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(y * x) + Float64(z * z))));
	else
		tmp = Float64(y * Float64(x + Float64(1.0 / Float64(Float64(y / z) / Float64(z * 3.0)))));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-21)
		tmp = (z * z) + ((z * z) + ((y * x) + (z * z)));
	else
		tmp = y * (x + (1.0 / ((y / z) / (z * 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[y, 2e-21], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(y * x), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x + N[(1.0 / N[(N[(y / z), $MachinePrecision] / N[(z * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999982e-21

   1. Initial program 97.8%

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

   if 1.99999999999999982e-21 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf 96.7%

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

   5. Simplified 96.7%

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

      1. unpow2 96.7%

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

      2. associate-/l* 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. associate-*l/ 99.9%

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

       2. clear-num 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-21}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(y \cdot x + z \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + \frac{1}{\frac{\frac{y}{z}}{z \cdot 3}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \cdot x \leq -\infty:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x + z \cdot z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= (* y x) (- INFINITY)) (* y x) (+ (* y x) (* z z))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((y * x) <= -((double) INFINITY)) {
		tmp = y * x;
	} else {
		tmp = (y * x) + (z * z);
	}
	return tmp;
}

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double tmp;
	if ((y * x) <= -Double.POSITIVE_INFINITY) {
		tmp = y * x;
	} else {
		tmp = (y * x) + (z * z);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (y * x) <= -math.inf:
		tmp = y * x
	else:
		tmp = (y * x) + (z * z)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (Float64(y * x) <= Float64(-Inf))
		tmp = Float64(y * x);
	else
		tmp = Float64(Float64(y * x) + Float64(z * z));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y * x) <= -Inf)
		tmp = y * x;
	else
		tmp = (y * x) + (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[N[(y * x), $MachinePrecision], (-Infinity)], N[(y * x), $MachinePrecision], N[(N[(y * x), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -inf.0

   1. Initial program 46.7%

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

   3. Taylor expanded in y around inf 73.3%

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

   5. Simplified 73.3%

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

   6. Taylor expanded in y around inf 73.3%

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

   if -inf.0 < (*.f64 x y)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 77.7%

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

   4. Taylor expanded in x around inf 77.1%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq -\infty:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x + z \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* y (+ x (/ 1.0 (/ (/ y z) (* z 3.0))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return y * (x + (1.0 / ((y / z) / (z * 3.0))));
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * (x + (1.0d0 / ((y / z) / (z * 3.0d0))))
end function

Java

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return y * (x + (1.0 / ((y / z) / (z * 3.0))))

Julia

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

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = y * (x + (1.0 / ((y / z) / (z * 3.0))));
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y * N[(x + N[(1.0 / N[(N[(y / z), $MachinePrecision] / N[(z * 3.0), $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. Add Preprocessing

3. Taylor expanded in y around inf 93.5%

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

5. Simplified 93.5%

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

   1. unpow2 93.5%

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

   2. associate-/l* 95.1%

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

7. Applied egg-rr 95.1%

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

   1. clear-num 95.1%

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

   2. un-div-inv 95.1%

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

9. Applied egg-rr 95.1%

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

    1. associate-*l/ 95.1%

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

    2. clear-num 95.1%

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

11. Applied egg-rr 95.1%

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

Alternative 5: 93.7% accurate, 1.4× speedup

Math

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

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* y (+ x (* 3.0 (* z (/ z y))))))

C

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

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * (x + (3.0d0 * (z * (z / y))))
end function

Java

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

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return y * (x + (3.0 * (z * (z / y))))

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(y * Float64(x + Float64(3.0 * Float64(z * Float64(z / y)))))
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = y * (x + (3.0 * (z * (z / y))));
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y * N[(x + N[(3.0 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

3. Taylor expanded in y around inf 93.5%

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

5. Simplified 93.5%

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

   1. unpow2 93.5%

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

   2. associate-/l* 95.1%

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

7. Applied egg-rr 95.1%

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

8. Final simplification 95.1%

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

Alternative 6: 52.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ y \cdot x \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z) :precision binary64 (* y x))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return y * x;
}

Fortran

NOTE: x, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * x
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	return y * x;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	return y * x

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(y * x)
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = y * x;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(y * x), $MachinePrecision]

Derivation

1. Initial program 96.7%

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

3. Taylor expanded in y around inf 93.5%

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

5. Simplified 93.5%

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

6. Taylor expanded in y around inf 51.4%

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

7. Final simplification 51.4%

   \[\leadsto y \cdot x \]
8. 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 2024106 
(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)))