Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, B

Percentage Accurate: 99.8% → 99.8%

Time: 5.7s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - 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 * 3.0d0) * y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))

C

double code(double x, double y, double z) {
	return ((x * 3.0) * y) - 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 * 3.0d0) * y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y, double z) {
	return fma((x * 3.0), y, -z);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Step-by-step derivation

   1. fma-neg 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-6} \lor \neg \left(x \leq 8 \cdot 10^{-32}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.7e-6) (not (<= x 8e-32))) (* 3.0 (* x y)) (- z)))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.7e-6) || !(x <= 8e-32)) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

NOTE: x and y 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 ((x <= (-1.7d-6)) .or. (.not. (x <= 8d-32))) then
        tmp = 3.0d0 * (x * y)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.7e-6) || !(x <= 8e-32)) {
		tmp = 3.0 * (x * y);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if (x <= -1.7e-6) or not (x <= 8e-32):
		tmp = 3.0 * (x * y)
	else:
		tmp = -z
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.7e-6) || !(x <= 8e-32))
		tmp = Float64(3.0 * Float64(x * y));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.7e-6) || ~((x <= 8e-32)))
		tmp = 3.0 * (x * y);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[x, -1.7e-6], N[Not[LessEqual[x, 8e-32]], $MachinePrecision]], N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000003e-6 or 8.00000000000000045e-32 < x

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 53.1%

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

      2. associate-*r* 53.1%

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

      3. fma-neg 53.1%

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

      4. add-sqr-sqrt 24.7%

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

      5. sqrt-unprod 43.1%

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

      6. sqr-neg 43.1%

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

      7. sqrt-unprod 21.1%

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

      8. add-sqr-sqrt 39.8%

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

   3. Applied egg-rr 39.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot 3\right) \cdot \sqrt{y}, \sqrt{y}, z\right)} \]
   4. Step-by-step derivation

      1. fma-udef 39.8%

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

      2. associate-*r* 39.8%

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

      3. add-sqr-sqrt 72.1%

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

      4. associate-*r* 72.1%

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

      5. flip3-+ 14.8%

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

      6. *-commutative 14.8%

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

      7. associate-*r* 14.7%

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

      8. *-commutative 14.7%

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

      9. unpow-prod-down 14.7%

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

      10. pow3 14.7%

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

      11. metadata-eval 14.7%

          \[\leadsto \frac{\left(\left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot x\right)\right) \cdot \color{blue}{27} + {z}^{3}}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

      12. metadata-eval 14.7%

          \[\leadsto \frac{\left(\left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot x\right)\right) \cdot \color{blue}{\left(9 \cdot 3\right)} + {z}^{3}}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

      13. metadata-eval 14.7%

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

      14. fma-def 14.7%

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

      15. pow3 14.7%

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

      16. *-commutative 14.7%

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

      17. metadata-eval 14.7%

          \[\leadsto \frac{\mathsf{fma}\left({\left(x \cdot y\right)}^{3}, \color{blue}{9} \cdot 3, {z}^{3}\right)}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

      18. metadata-eval 14.7%

          \[\leadsto \frac{\mathsf{fma}\left({\left(x \cdot y\right)}^{3}, \color{blue}{27}, {z}^{3}\right)}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

   5. Applied egg-rr 14.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot y\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(x \cdot y\right)}^{2}, 9, z \cdot z - x \cdot \left(\left(3 \cdot y\right) \cdot z\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 14.8%

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(y \cdot x\right)}}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(x \cdot y\right)}^{2}, 9, z \cdot z - x \cdot \left(\left(3 \cdot y\right) \cdot z\right)\right)} \]

      2. *-commutative 14.8%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\color{blue}{\left(y \cdot x\right)}}^{2}, 9, z \cdot z - x \cdot \left(\left(3 \cdot y\right) \cdot z\right)\right)} \]

      3. associate-*r* 14.8%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot z - \color{blue}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot z}\right)} \]

      4. distribute-rgt-out-- 14.8%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, \color{blue}{z \cdot \left(z - x \cdot \left(3 \cdot y\right)\right)}\right)} \]

      5. *-commutative 14.8%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - \color{blue}{\left(3 \cdot y\right) \cdot x}\right)\right)} \]

      6. associate-*r* 14.8%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - \color{blue}{3 \cdot \left(y \cdot x\right)}\right)\right)} \]

   7. Simplified 14.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - 3 \cdot \left(y \cdot x\right)\right)\right)}} \]

   8. Taylor expanded in y around inf 72.6%

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

   if -1.70000000000000003e-6 < x < 8.00000000000000045e-32

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]

   2. Taylor expanded in x around 0 76.0%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. mul-1-neg 76.0%

         \[\leadsto \color{blue}{-z} \]

   4. Simplified 76.0%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-6} \lor \neg \left(x \leq 8 \cdot 10^{-32}\right):\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 3: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -1.8e-6) (* (* x 3.0) y) (if (<= x 3.1e-29) (- z) (* 3.0 (* x y)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e-6) {
		tmp = (x * 3.0) * y;
	} else if (x <= 3.1e-29) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Fortran

NOTE: x and y 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 (x <= (-1.8d-6)) then
        tmp = (x * 3.0d0) * y
    else if (x <= 3.1d-29) then
        tmp = -z
    else
        tmp = 3.0d0 * (x * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.8e-6) {
		tmp = (x * 3.0) * y;
	} else if (x <= 3.1e-29) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	tmp = 0
	if x <= -1.8e-6:
		tmp = (x * 3.0) * y
	elif x <= 3.1e-29:
		tmp = -z
	else:
		tmp = 3.0 * (x * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	tmp = 0.0
	if (x <= -1.8e-6)
		tmp = Float64(Float64(x * 3.0) * y);
	elseif (x <= 3.1e-29)
		tmp = Float64(-z);
	else
		tmp = Float64(3.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.8e-6)
		tmp = (x * 3.0) * y;
	elseif (x <= 3.1e-29)
		tmp = -z;
	else
		tmp = 3.0 * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[LessEqual[x, -1.8e-6], N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 3.1e-29], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.79999999999999992e-6

   1. Initial program 99.8%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 53.6%

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

      2. associate-*r* 53.7%

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

      3. fma-neg 53.7%

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

      4. add-sqr-sqrt 26.9%

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

      5. sqrt-unprod 50.5%

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

      6. sqr-neg 50.5%

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

      7. sqrt-unprod 23.6%

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

      8. add-sqr-sqrt 46.2%

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

   3. Applied egg-rr 46.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot 3\right) \cdot \sqrt{y}, \sqrt{y}, z\right)} \]
   4. Step-by-step derivation

      1. fma-udef 46.2%

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

      2. associate-*r* 46.2%

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

      3. add-sqr-sqrt 80.1%

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

      4. associate-*r* 80.1%

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

      5. flip3-+ 14.1%

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

      6. *-commutative 14.1%

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

      7. associate-*r* 14.1%

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

      8. *-commutative 14.1%

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

      9. unpow-prod-down 14.0%

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

      10. pow3 14.0%

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

      11. metadata-eval 14.0%

          \[\leadsto \frac{\left(\left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot x\right)\right) \cdot \color{blue}{27} + {z}^{3}}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

      12. metadata-eval 14.0%

          \[\leadsto \frac{\left(\left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot x\right)\right) \cdot \color{blue}{\left(9 \cdot 3\right)} + {z}^{3}}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

      13. metadata-eval 14.0%

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

      14. fma-def 14.0%

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

      15. pow3 14.0%

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

      16. *-commutative 14.0%

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

      17. metadata-eval 14.0%

          \[\leadsto \frac{\mathsf{fma}\left({\left(x \cdot y\right)}^{3}, \color{blue}{9} \cdot 3, {z}^{3}\right)}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

      18. metadata-eval 14.0%

          \[\leadsto \frac{\mathsf{fma}\left({\left(x \cdot y\right)}^{3}, \color{blue}{27}, {z}^{3}\right)}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

   5. Applied egg-rr 14.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot y\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(x \cdot y\right)}^{2}, 9, z \cdot z - x \cdot \left(\left(3 \cdot y\right) \cdot z\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 14.0%

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(y \cdot x\right)}}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(x \cdot y\right)}^{2}, 9, z \cdot z - x \cdot \left(\left(3 \cdot y\right) \cdot z\right)\right)} \]

      2. *-commutative 14.0%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\color{blue}{\left(y \cdot x\right)}}^{2}, 9, z \cdot z - x \cdot \left(\left(3 \cdot y\right) \cdot z\right)\right)} \]

      3. associate-*r* 14.0%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot z - \color{blue}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot z}\right)} \]

      4. distribute-rgt-out-- 14.0%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, \color{blue}{z \cdot \left(z - x \cdot \left(3 \cdot y\right)\right)}\right)} \]

      5. *-commutative 14.0%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - \color{blue}{\left(3 \cdot y\right) \cdot x}\right)\right)} \]

      6. associate-*r* 14.0%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - \color{blue}{3 \cdot \left(y \cdot x\right)}\right)\right)} \]

   7. Simplified 14.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - 3 \cdot \left(y \cdot x\right)\right)\right)}} \]

   8. Taylor expanded in y around inf 9.3%

      \[\leadsto \frac{\color{blue}{27 \cdot \left({y}^{3} \cdot {x}^{3}\right)}}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - 3 \cdot \left(y \cdot x\right)\right)\right)} \]
   9. Step-by-step derivation

      1. cube-prod 14.3%

         \[\leadsto \frac{27 \cdot \color{blue}{{\left(y \cdot x\right)}^{3}}}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - 3 \cdot \left(y \cdot x\right)\right)\right)} \]

   10. Simplified 14.3%

       \[\leadsto \frac{\color{blue}{27 \cdot {\left(y \cdot x\right)}^{3}}}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - 3 \cdot \left(y \cdot x\right)\right)\right)} \]

   11. Taylor expanded in y around inf 80.6%

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

       1. *-commutative 80.6%

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

       2. associate-*r* 80.6%

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

   13. Simplified 80.6%

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

   if -1.79999999999999992e-6 < x < 3.10000000000000026e-29

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]

   2. Taylor expanded in x around 0 76.0%

      \[\leadsto \color{blue}{-1 \cdot z} \]
   3. Step-by-step derivation

      1. mul-1-neg 76.0%

         \[\leadsto \color{blue}{-z} \]

   4. Simplified 76.0%

      \[\leadsto \color{blue}{-z} \]

   if 3.10000000000000026e-29 < x

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 52.6%

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

      2. associate-*r* 52.5%

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

      3. fma-neg 52.5%

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

      4. add-sqr-sqrt 22.8%

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

      5. sqrt-unprod 36.8%

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

      6. sqr-neg 36.8%

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

      7. sqrt-unprod 18.9%

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

      8. add-sqr-sqrt 34.3%

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

   3. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot 3\right) \cdot \sqrt{y}, \sqrt{y}, z\right)} \]
   4. Step-by-step derivation

      1. fma-udef 34.3%

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

      2. associate-*r* 34.4%

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

      3. add-sqr-sqrt 65.3%

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

      4. associate-*r* 65.3%

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

      5. flip3-+ 15.4%

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

      6. *-commutative 15.4%

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

      7. associate-*r* 15.3%

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

      8. *-commutative 15.3%

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

      9. unpow-prod-down 15.3%

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

      10. pow3 15.3%

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

      11. metadata-eval 15.3%

          \[\leadsto \frac{\left(\left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot x\right)\right) \cdot \color{blue}{27} + {z}^{3}}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

      12. metadata-eval 15.3%

          \[\leadsto \frac{\left(\left(\left(y \cdot x\right) \cdot \left(y \cdot x\right)\right) \cdot \left(y \cdot x\right)\right) \cdot \color{blue}{\left(9 \cdot 3\right)} + {z}^{3}}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

      13. metadata-eval 15.3%

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

      14. fma-def 15.3%

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

      15. pow3 15.3%

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

      16. *-commutative 15.3%

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

      17. metadata-eval 15.3%

          \[\leadsto \frac{\mathsf{fma}\left({\left(x \cdot y\right)}^{3}, \color{blue}{9} \cdot 3, {z}^{3}\right)}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

      18. metadata-eval 15.3%

          \[\leadsto \frac{\mathsf{fma}\left({\left(x \cdot y\right)}^{3}, \color{blue}{27}, {z}^{3}\right)}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot \left(x \cdot \left(3 \cdot y\right)\right) + \left(z \cdot z - \left(x \cdot \left(3 \cdot y\right)\right) \cdot z\right)} \]

   5. Applied egg-rr 15.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(x \cdot y\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(x \cdot y\right)}^{2}, 9, z \cdot z - x \cdot \left(\left(3 \cdot y\right) \cdot z\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 15.4%

         \[\leadsto \frac{\mathsf{fma}\left({\color{blue}{\left(y \cdot x\right)}}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(x \cdot y\right)}^{2}, 9, z \cdot z - x \cdot \left(\left(3 \cdot y\right) \cdot z\right)\right)} \]

      2. *-commutative 15.4%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\color{blue}{\left(y \cdot x\right)}}^{2}, 9, z \cdot z - x \cdot \left(\left(3 \cdot y\right) \cdot z\right)\right)} \]

      3. associate-*r* 15.4%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot z - \color{blue}{\left(x \cdot \left(3 \cdot y\right)\right) \cdot z}\right)} \]

      4. distribute-rgt-out-- 15.4%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, \color{blue}{z \cdot \left(z - x \cdot \left(3 \cdot y\right)\right)}\right)} \]

      5. *-commutative 15.4%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - \color{blue}{\left(3 \cdot y\right) \cdot x}\right)\right)} \]

      6. associate-*r* 15.4%

         \[\leadsto \frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - \color{blue}{3 \cdot \left(y \cdot x\right)}\right)\right)} \]

   7. Simplified 15.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(y \cdot x\right)}^{3}, 27, {z}^{3}\right)}{\mathsf{fma}\left({\left(y \cdot x\right)}^{2}, 9, z \cdot \left(z - 3 \cdot \left(y \cdot x\right)\right)\right)}} \]

   8. Taylor expanded in y around inf 65.8%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-6}:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-29}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y 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 = (3.0d0 * (x * y)) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]

2. Taylor expanded in x around 0 99.8%

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

3. Final simplification 99.8%

   \[\leadsto 3 \cdot \left(x \cdot y\right) - z \]

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y 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 = (x * (3.0d0 * y)) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Step-by-step derivation

   1. associate-*l* 99.9%

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

   2. fma-neg 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 3 \cdot y, -z\right)} \]
4. Step-by-step derivation

   1. fma-neg 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

   \[\leadsto x \cdot \left(3 \cdot y\right) - z \]

Alternative 6: 49.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y 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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]

2. Taylor expanded in x around 0 50.3%

   \[\leadsto \color{blue}{-1 \cdot z} \]
3. Step-by-step derivation

   1. mul-1-neg 50.3%

      \[\leadsto \color{blue}{-z} \]

4. Simplified 50.3%

   \[\leadsto \color{blue}{-z} \]

5. Final simplification 50.3%

   \[\leadsto -z \]

Alternative 7: 2.3% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x and y 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 = z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Step-by-step derivation

   1. add-sqr-sqrt 49.4%

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

   2. associate-*r* 49.4%

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

   3. fma-neg 49.4%

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

   4. add-sqr-sqrt 22.5%

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

   5. sqrt-unprod 33.1%

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

   6. sqr-neg 33.1%

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

   7. sqrt-unprod 14.5%

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

   8. add-sqr-sqrt 27.0%

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

3. Applied egg-rr 27.0%

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

4. Taylor expanded in x around 0 2.5%

   \[\leadsto \color{blue}{z} \]

5. Final simplification 2.5%

   \[\leadsto z \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (* x (* 3.0 y)) z))

C

double code(double x, double y, double z) {
	return (x * (3.0 * y)) - 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 * (3.0d0 * y)) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023238 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))