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

Percentage Accurate: 99.8% → 99.8%

Time: 6.8s

Alternatives: 5

Speedup: 0.1×

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

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

FPCore

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

C

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

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	return Float64(-fma(x, Float64(y * -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[(x * N[(y * -3.0), $MachinePrecision] + z), $MachinePrecision])

Derivation

1. Initial program 99.5%

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

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. cancel-sign-sub 99.5%

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

   4. distribute-lft-neg-out 99.5%

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

   5. unsub-neg 99.5%

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

   6. distribute-neg-out 99.5%

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

   7. +-commutative 99.5%

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

   8. distribute-lft-neg-out 99.5%

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

   9. distribute-lft-neg-in 99.5%

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

   10. distribute-rgt-neg-out 99.5%

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

   11. associate-*l* 99.4%

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

   12. fma-def 99.4%

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

   13. neg-mul-1 99.4%

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

   14. associate-*r* 99.4%

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

   15. *-commutative 99.4%

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

   16. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+137} \lor \neg \left(z \leq -5.2 \cdot 10^{+75} \lor \neg \left(z \leq -2.8 \cdot 10^{-16}\right) \land z \leq 3.35 \cdot 10^{+53}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(--3\right)\\ \end{array} \end{array} \]

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 -2.4e+137)
         (not
          (or (<= z -5.2e+75) (and (not (<= z -2.8e-16)) (<= z 3.35e+53)))))
   (- z)
   (* (* x y) (- -3.0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.4e+137) || !((z <= -5.2e+75) || (!(z <= -2.8e-16) && (z <= 3.35e+53)))) {
		tmp = -z;
	} else {
		tmp = (x * y) * -(-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 <= (-2.4d+137)) .or. (.not. (z <= (-5.2d+75)) .or. (.not. (z <= (-2.8d-16))) .and. (z <= 3.35d+53))) then
        tmp = -z
    else
        tmp = (x * y) * -(-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 <= -2.4e+137) || !((z <= -5.2e+75) || (!(z <= -2.8e-16) && (z <= 3.35e+53)))) {
		tmp = -z;
	} else {
		tmp = (x * y) * -(-3.0);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z <= -2.4e+137) or not ((z <= -5.2e+75) or (not (z <= -2.8e-16) and (z <= 3.35e+53))):
		tmp = -z
	else:
		tmp = (x * y) * -(-3.0)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.4e+137) || !((z <= -5.2e+75) || (!(z <= -2.8e-16) && (z <= 3.35e+53))))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * y) * Float64(-(-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 <= -2.4e+137) || ~(((z <= -5.2e+75) || (~((z <= -2.8e-16)) && (z <= 3.35e+53)))))
		tmp = -z;
	else
		tmp = (x * y) * -(-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, -2.4e+137], N[Not[Or[LessEqual[z, -5.2e+75], And[N[Not[LessEqual[z, -2.8e-16]], $MachinePrecision], LessEqual[z, 3.35e+53]]]], $MachinePrecision]], (-z), N[(N[(x * y), $MachinePrecision] * (--3.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.39999999999999983e137 or -5.1999999999999997e75 < z < -2.8000000000000001e-16 or 3.3499999999999999e53 < z

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-neg-out 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. associate-*l* 100.0%

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

      12. fma-def 100.0%

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

      13. neg-mul-1 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 85.2%

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

   if -2.39999999999999983e137 < z < -5.1999999999999997e75 or -2.8000000000000001e-16 < z < 3.3499999999999999e53

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

      2. +-commutative 99.1%

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

      3. cancel-sign-sub 99.1%

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

      4. distribute-lft-neg-out 99.1%

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

      5. unsub-neg 99.1%

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

      6. distribute-neg-out 99.1%

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

      7. +-commutative 99.1%

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

      8. distribute-lft-neg-out 99.1%

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

      9. distribute-lft-neg-in 99.1%

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

      10. distribute-rgt-neg-out 99.1%

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

      11. associate-*l* 99.1%

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

      12. fma-def 99.1%

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

      13. neg-mul-1 99.1%

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

      14. associate-*r* 99.1%

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

      15. *-commutative 99.1%

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

      16. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around inf 72.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+137} \lor \neg \left(z \leq -5.2 \cdot 10^{+75} \lor \neg \left(z \leq -2.8 \cdot 10^{-16}\right) \land z \leq 3.35 \cdot 10^{+53}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \left(--3\right)\\ \end{array} \]

Alternative 3: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+137} \lor \neg \left(z \leq -1.35 \cdot 10^{+75}\right) \land \left(z \leq -2.05 \cdot 10^{-16} \lor \neg \left(z \leq 5.8 \cdot 10^{+52}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -3\right) \cdot \left(-x\right)\\ \end{array} \end{array} \]

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 -2.4e+137)
         (and (not (<= z -1.35e+75))
              (or (<= z -2.05e-16) (not (<= z 5.8e+52)))))
   (- z)
   (* (* y -3.0) (- x))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.4e+137) || (!(z <= -1.35e+75) && ((z <= -2.05e-16) || !(z <= 5.8e+52)))) {
		tmp = -z;
	} else {
		tmp = (y * -3.0) * -x;
	}
	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 <= (-2.4d+137)) .or. (.not. (z <= (-1.35d+75))) .and. (z <= (-2.05d-16)) .or. (.not. (z <= 5.8d+52))) then
        tmp = -z
    else
        tmp = (y * (-3.0d0)) * -x
    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 <= -2.4e+137) || (!(z <= -1.35e+75) && ((z <= -2.05e-16) || !(z <= 5.8e+52)))) {
		tmp = -z;
	} else {
		tmp = (y * -3.0) * -x;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (z <= -2.4e+137) or (not (z <= -1.35e+75) and ((z <= -2.05e-16) or not (z <= 5.8e+52))):
		tmp = -z
	else:
		tmp = (y * -3.0) * -x
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.4e+137) || (!(z <= -1.35e+75) && ((z <= -2.05e-16) || !(z <= 5.8e+52))))
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(y * -3.0) * Float64(-x));
	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 <= -2.4e+137) || (~((z <= -1.35e+75)) && ((z <= -2.05e-16) || ~((z <= 5.8e+52)))))
		tmp = -z;
	else
		tmp = (y * -3.0) * -x;
	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, -2.4e+137], And[N[Not[LessEqual[z, -1.35e+75]], $MachinePrecision], Or[LessEqual[z, -2.05e-16], N[Not[LessEqual[z, 5.8e+52]], $MachinePrecision]]]], (-z), N[(N[(y * -3.0), $MachinePrecision] * (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.39999999999999983e137 or -1.34999999999999999e75 < z < -2.05000000000000003e-16 or 5.8e52 < z

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. cancel-sign-sub 100.0%

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

      4. distribute-lft-neg-out 100.0%

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

      5. unsub-neg 100.0%

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

      6. distribute-neg-out 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-neg-out 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

      11. associate-*l* 100.0%

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

      12. fma-def 100.0%

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

      13. neg-mul-1 100.0%

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

      14. associate-*r* 100.0%

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

      15. *-commutative 100.0%

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

      16. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 85.2%

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

   if -2.39999999999999983e137 < z < -1.34999999999999999e75 or -2.05000000000000003e-16 < z < 5.8e52

   1. Initial program 99.1%

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

      1. sub-neg 99.1%

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

      2. +-commutative 99.1%

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

      3. cancel-sign-sub 99.1%

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

      4. distribute-lft-neg-out 99.1%

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

      5. unsub-neg 99.1%

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

      6. distribute-neg-out 99.1%

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

      7. +-commutative 99.1%

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

      8. distribute-lft-neg-out 99.1%

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

      9. distribute-lft-neg-in 99.1%

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

      10. distribute-rgt-neg-out 99.1%

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

      11. associate-*l* 99.1%

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

      12. fma-def 99.1%

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

      13. neg-mul-1 99.1%

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

      14. associate-*r* 99.1%

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

      15. *-commutative 99.1%

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

      16. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-*r* 72.1%

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

      3. *-commutative 72.1%

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

   6. Simplified 72.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+137} \lor \neg \left(z \leq -1.35 \cdot 10^{+75}\right) \land \left(z \leq -2.05 \cdot 10^{-16} \lor \neg \left(z \leq 5.8 \cdot 10^{+52}\right)\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -3\right) \cdot \left(-x\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	return (x * (y * 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 = (x * (y * 3.0d0)) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*l* 99.4%

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

   4. *-commutative 99.4%

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

   5. *-commutative 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 5: 50.8% 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 99.5%

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

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. cancel-sign-sub 99.5%

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

   4. distribute-lft-neg-out 99.5%

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

   5. unsub-neg 99.5%

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

   6. distribute-neg-out 99.5%

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

   7. +-commutative 99.5%

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

   8. distribute-lft-neg-out 99.5%

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

   9. distribute-lft-neg-in 99.5%

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

   10. distribute-rgt-neg-out 99.5%

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

   11. associate-*l* 99.4%

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

   12. fma-def 99.4%

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

   13. neg-mul-1 99.4%

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

   14. associate-*r* 99.4%

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

   15. *-commutative 99.4%

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

   16. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 51.6%

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

5. Final simplification 51.6%

   \[\leadsto -z \]

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 2023336 
(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))