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

Percentage Accurate: 99.8% → 99.8%

Time: 4.3s

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, 1.0× speedup

Math

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

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))

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(x \cdot 3\right) \cdot y - z \]
2. Add Preprocessing

3. Final simplification 99.9%

   \[\leadsto y \cdot \left(x \cdot 3\right) - z \]
4. Add Preprocessing

Alternative 2: 70.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+72} \lor \neg \left(z \leq 2.9 \cdot 10^{-53}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.95e+72) (not (<= z 2.9e-53))) (- z) (* y (* x 3.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.95e+72) || !(z <= 2.9e-53)) {
		tmp = -z;
	} else {
		tmp = y * (x * 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 ((z <= (-1.95d+72)) .or. (.not. (z <= 2.9d-53))) then
        tmp = -z
    else
        tmp = y * (x * 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 ((z <= -1.95e+72) || !(z <= 2.9e-53)) {
		tmp = -z;
	} else {
		tmp = y * (x * 3.0);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z <= -1.95e+72) or not (z <= 2.9e-53):
		tmp = -z
	else:
		tmp = y * (x * 3.0)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.95e+72) || !(z <= 2.9e-53))
		tmp = Float64(-z);
	else
		tmp = Float64(y * Float64(x * 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 ((z <= -1.95e+72) || ~((z <= 2.9e-53)))
		tmp = -z;
	else
		tmp = y * (x * 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[Or[LessEqual[z, -1.95e+72], N[Not[LessEqual[z, 2.9e-53]], $MachinePrecision]], (-z), N[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.94999999999999996e72 or 2.8999999999999998e-53 < z

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 86.4%

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

      3. pow2 86.4%

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

      4. *-commutative 86.4%

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

      5. associate-*l* 86.4%

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

      6. unpow-prod-down 86.3%

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

      7. metadata-eval 86.3%

         \[\leadsto \sqrt{\color{blue}{9} \cdot {\left(y \cdot x\right)}^{2}} - z \]

   6. Applied egg-rr 86.3%

      \[\leadsto \color{blue}{\sqrt{9 \cdot {\left(y \cdot x\right)}^{2}}} - z \]

   7. Taylor expanded in y around 0 77.7%

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

      1. neg-mul-1 77.7%

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

   9. Simplified 77.7%

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

   if -1.94999999999999996e72 < z < 2.8999999999999998e-53

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 53.3%

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

      2. sqrt-unprod 53.2%

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

      3. pow2 53.2%

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

      4. *-commutative 53.2%

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

      5. associate-*l* 53.2%

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

      6. unpow-prod-down 53.3%

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

      7. metadata-eval 53.3%

         \[\leadsto \sqrt{\color{blue}{9} \cdot {\left(y \cdot x\right)}^{2}} - z \]

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\sqrt{9 \cdot {\left(y \cdot x\right)}^{2}}} - z \]

   7. Taylor expanded in z around inf 81.2%

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

   8. Taylor expanded in z around 0 77.6%

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

      1. associate-*r* 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+72} \lor \neg \left(z \leq 2.9 \cdot 10^{-53}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+72} \lor \neg \left(z \leq 1.15 \cdot 10^{-52}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.28e+72) (not (<= z 1.15e-52))) (- z) (* x (* 3.0 y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.28e+72) || !(z <= 1.15e-52)) {
		tmp = -z;
	} else {
		tmp = x * (3.0 * y);
	}
	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 ((z <= (-1.28d+72)) .or. (.not. (z <= 1.15d-52))) then
        tmp = -z
    else
        tmp = x * (3.0d0 * y)
    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 ((z <= -1.28e+72) || !(z <= 1.15e-52)) {
		tmp = -z;
	} else {
		tmp = x * (3.0 * y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z <= -1.28e+72) or not (z <= 1.15e-52):
		tmp = -z
	else:
		tmp = x * (3.0 * y)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.28e+72) || !(z <= 1.15e-52))
		tmp = Float64(-z);
	else
		tmp = Float64(x * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.28e+72) || ~((z <= 1.15e-52)))
		tmp = -z;
	else
		tmp = x * (3.0 * y);
	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[Or[LessEqual[z, -1.28e+72], N[Not[LessEqual[z, 1.15e-52]], $MachinePrecision]], (-z), N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.28000000000000009e72 or 1.14999999999999997e-52 < z

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 86.4%

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

      3. pow2 86.4%

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

      4. *-commutative 86.4%

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

      5. associate-*l* 86.4%

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

      6. unpow-prod-down 86.3%

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

      7. metadata-eval 86.3%

         \[\leadsto \sqrt{\color{blue}{9} \cdot {\left(y \cdot x\right)}^{2}} - z \]

   6. Applied egg-rr 86.3%

      \[\leadsto \color{blue}{\sqrt{9 \cdot {\left(y \cdot x\right)}^{2}}} - z \]

   7. Taylor expanded in y around 0 77.7%

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

      1. neg-mul-1 77.7%

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

   9. Simplified 77.7%

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

   if -1.28000000000000009e72 < z < 1.14999999999999997e-52

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 53.3%

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

      2. sqrt-unprod 53.2%

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

      3. pow2 53.2%

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

      4. *-commutative 53.2%

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

      5. associate-*l* 53.2%

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

      6. unpow-prod-down 53.3%

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

      7. metadata-eval 53.3%

         \[\leadsto \sqrt{\color{blue}{9} \cdot {\left(y \cdot x\right)}^{2}} - z \]

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\sqrt{9 \cdot {\left(y \cdot x\right)}^{2}}} - z \]

   7. Taylor expanded in z around inf 81.2%

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

   8. Taylor expanded in z around 0 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*r* 77.7%

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

      3. *-commutative 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+72} \lor \neg \left(z \leq 1.15 \cdot 10^{-52}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+72} \lor \neg \left(z \leq 1.06 \cdot 10^{-52}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.28e+72) (not (<= z 1.06e-52))) (- z) (* 3.0 (* x y))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.28e+72) || !(z <= 1.06e-52)) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	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 ((z <= (-1.28d+72)) .or. (.not. (z <= 1.06d-52))) then
        tmp = -z
    else
        tmp = 3.0d0 * (x * y)
    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 ((z <= -1.28e+72) || !(z <= 1.06e-52)) {
		tmp = -z;
	} else {
		tmp = 3.0 * (x * y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z <= -1.28e+72) or not (z <= 1.06e-52):
		tmp = -z
	else:
		tmp = 3.0 * (x * y)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.28e+72) || !(z <= 1.06e-52))
		tmp = Float64(-z);
	else
		tmp = Float64(3.0 * Float64(x * y));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.28e+72) || ~((z <= 1.06e-52)))
		tmp = -z;
	else
		tmp = 3.0 * (x * y);
	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[Or[LessEqual[z, -1.28e+72], N[Not[LessEqual[z, 1.06e-52]], $MachinePrecision]], (-z), N[(3.0 * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.28000000000000009e72 or 1.06e-52 < z

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 86.4%

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

      3. pow2 86.4%

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

      4. *-commutative 86.4%

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

      5. associate-*l* 86.4%

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

      6. unpow-prod-down 86.3%

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

      7. metadata-eval 86.3%

         \[\leadsto \sqrt{\color{blue}{9} \cdot {\left(y \cdot x\right)}^{2}} - z \]

   6. Applied egg-rr 86.3%

      \[\leadsto \color{blue}{\sqrt{9 \cdot {\left(y \cdot x\right)}^{2}}} - z \]

   7. Taylor expanded in y around 0 77.7%

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

      1. neg-mul-1 77.7%

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

   9. Simplified 77.7%

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

   if -1.28000000000000009e72 < z < 1.06e-52

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

      1. add-sqr-sqrt 53.3%

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

      2. sqrt-unprod 53.2%

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

      3. pow2 53.2%

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

      4. *-commutative 53.2%

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

      5. associate-*l* 53.2%

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

      6. unpow-prod-down 53.3%

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

      7. metadata-eval 53.3%

         \[\leadsto \sqrt{\color{blue}{9} \cdot {\left(y \cdot x\right)}^{2}} - z \]

   6. Applied egg-rr 53.3%

      \[\leadsto \color{blue}{\sqrt{9 \cdot {\left(y \cdot x\right)}^{2}}} - z \]

   7. Taylor expanded in z around inf 81.2%

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

   8. Taylor expanded in z around 0 77.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+72} \lor \neg \left(z \leq 1.06 \cdot 10^{-52}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z 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 && y < z);
double code(double x, double y, double z) {
	return (x * (3.0 * y)) - z;
}

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, and z 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 99.9%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x, y, and z 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 && y < z);
double code(double x, double y, double z) {
	return (3.0 * (x * y)) - z;
}

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, and z 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 99.9%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.8%

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

Alternative 7: 50.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

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 = -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 99.9%

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

3. Simplified 99.9%

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

   1. add-sqr-sqrt 54.1%

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

   2. sqrt-unprod 69.9%

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

   3. pow2 69.9%

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

   4. *-commutative 69.9%

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

   5. associate-*l* 69.9%

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

   6. unpow-prod-down 69.9%

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

   7. metadata-eval 69.9%

      \[\leadsto \sqrt{\color{blue}{9} \cdot {\left(y \cdot x\right)}^{2}} - z \]

6. Applied egg-rr 69.9%

   \[\leadsto \color{blue}{\sqrt{9 \cdot {\left(y \cdot x\right)}^{2}}} - z \]

7. Taylor expanded in y around 0 51.0%

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

   1. neg-mul-1 51.0%

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

9. Simplified 51.0%

   \[\leadsto \color{blue}{-z} \]
10. Add Preprocessing

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 2024086 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

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

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