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

Percentage Accurate: 99.8% → 99.8%

Time: 6.5s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l* 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-neg 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y \cdot \left(3 \cdot x\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 10000000000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* 3.0 x))))
   (if (<= t_0 -5e+70)
     t_0
     (if (<= t_0 10000000000000.0) (- z) (* (* 3.0 y) x)))))

C

assert(x < y);
double code(double x, double y, double z) {
	double t_0 = y * (3.0 * x);
	double tmp;
	if (t_0 <= -5e+70) {
		tmp = t_0;
	} else if (t_0 <= 10000000000000.0) {
		tmp = -z;
	} else {
		tmp = (3.0 * y) * x;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = y * (3.0d0 * x)
    if (t_0 <= (-5d+70)) then
        tmp = t_0
    else if (t_0 <= 10000000000000.0d0) then
        tmp = -z
    else
        tmp = (3.0d0 * y) * x
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z) {
	double t_0 = y * (3.0 * x);
	double tmp;
	if (t_0 <= -5e+70) {
		tmp = t_0;
	} else if (t_0 <= 10000000000000.0) {
		tmp = -z;
	} else {
		tmp = (3.0 * y) * x;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z):
	t_0 = y * (3.0 * x)
	tmp = 0
	if t_0 <= -5e+70:
		tmp = t_0
	elif t_0 <= 10000000000000.0:
		tmp = -z
	else:
		tmp = (3.0 * y) * x
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z)
	t_0 = Float64(y * Float64(3.0 * x))
	tmp = 0.0
	if (t_0 <= -5e+70)
		tmp = t_0;
	elseif (t_0 <= 10000000000000.0)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(3.0 * y) * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (3.0 * x);
	tmp = 0.0;
	if (t_0 <= -5e+70)
		tmp = t_0;
	elseif (t_0 <= 10000000000000.0)
		tmp = -z;
	else
		tmp = (3.0 * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(3.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+70], t$95$0, If[LessEqual[t$95$0, 10000000000000.0], (-z), N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x 3) y) < -5.0000000000000002e70

   1. Initial program 99.8%

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

      1. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r* 99.8%

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

      2. add-sqr-sqrt 57.6%

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

      3. associate-*r* 57.7%

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

      4. fma-neg 57.7%

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

      5. add-sqr-sqrt 27.8%

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

      6. sqrt-unprod 45.7%

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

      7. sqr-neg 45.7%

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

      8. sqrt-unprod 19.9%

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

      9. add-sqr-sqrt 47.7%

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

   5. Applied egg-rr 47.7%

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

   6. Taylor expanded in x around inf 88.3%

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

      1. rem-cube-cbrt 87.1%

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

      2. associate-*r* 87.2%

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

      3. *-commutative 87.2%

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

      4. associate-*l* 87.1%

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

   8. Applied egg-rr 87.1%

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

      1. rem-cube-cbrt 88.2%

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

      2. associate-*r* 88.4%

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

   10. Applied egg-rr 88.4%

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

   if -5.0000000000000002e70 < (*.f64 (*.f64 x 3) y) < 1e13

   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. Taylor expanded in x around 0 77.7%

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

      1. mul-1-neg 77.7%

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

   6. Simplified 77.7%

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

   if 1e13 < (*.f64 (*.f64 x 3) y)

   1. Initial program 98.2%

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

      1. associate-*l* 98.2%

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

   3. Simplified 98.2%

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

      1. associate-*r* 98.2%

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

      2. add-sqr-sqrt 55.7%

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

      3. associate-*r* 55.6%

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

      4. fma-neg 55.6%

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

      5. add-sqr-sqrt 26.9%

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

      6. sqrt-unprod 41.5%

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

      7. sqr-neg 41.5%

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

      8. sqrt-unprod 21.2%

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

      9. add-sqr-sqrt 43.1%

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

   5. Applied egg-rr 43.1%

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

   6. Taylor expanded in x around inf 81.1%

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

      1. associate-*r* 79.6%

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

      2. *-commutative 79.6%

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

      3. associate-*r* 79.5%

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

   8. Simplified 79.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(3 \cdot x\right) \leq -5 \cdot 10^{+70}:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{elif}\;y \cdot \left(3 \cdot x\right) \leq 10000000000000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \]

Alternative 3: 68.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.4e+137)
         (not
          (or (<= z -5.2e+75) (and (not (<= z -2.8e-16)) (<= z 3.35e+53)))))
   (- z)
   (* 3.0 (* y x))))

C

assert(x < y);
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 = 3.0 * (y * x);
	}
	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 ((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 = 3.0d0 * (y * x)
    end if
    code = tmp
end function

Java

assert x < y;
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 = 3.0 * (y * x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
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 = 3.0 * (y * x)
	return tmp

Julia

x, y = sort([x, y])
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(3.0 * Float64(y * x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
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 = 3.0 * (y * x);
	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[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[(3.0 * N[(y * x), $MachinePrecision]), $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. 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. Taylor expanded in x around 0 85.2%

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

      1. mul-1-neg 85.2%

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

   6. Simplified 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. associate-*l* 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r* 99.1%

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

      2. add-sqr-sqrt 51.3%

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

      3. associate-*r* 51.3%

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

      4. fma-neg 51.4%

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

      5. add-sqr-sqrt 26.0%

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

      6. sqrt-unprod 42.7%

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

      7. sqr-neg 42.7%

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

      8. sqrt-unprod 18.1%

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

      9. add-sqr-sqrt 39.7%

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

   5. Applied egg-rr 39.7%

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

   6. 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%

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

Alternative 4: 68.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.4e+137)
         (not
          (or (<= z -1.35e+75) (and (not (<= z -2.05e-16)) (<= z 5.8e+52)))))
   (- z)
   (* (* 3.0 y) x)))

C

assert(x < y);
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 = (3.0 * y) * x;
	}
	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 ((z <= (-2.4d+137)) .or. (.not. (z <= (-1.35d+75)) .or. (.not. (z <= (-2.05d-16))) .and. (z <= 5.8d+52))) then
        tmp = -z
    else
        tmp = (3.0d0 * y) * x
    end if
    code = tmp
end function

Java

assert x < y;
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 = (3.0 * y) * x;
	}
	return tmp;
}

Python

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

Julia

x, y = sort([x, y])
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(3.0 * y) * x);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
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 = (3.0 * y) * x;
	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[z, -2.4e+137], N[Not[Or[LessEqual[z, -1.35e+75], And[N[Not[LessEqual[z, -2.05e-16]], $MachinePrecision], LessEqual[z, 5.8e+52]]]], $MachinePrecision]], (-z), N[(N[(3.0 * y), $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. 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. Taylor expanded in x around 0 85.2%

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

      1. mul-1-neg 85.2%

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

   6. Simplified 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. associate-*l* 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r* 99.1%

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

      2. add-sqr-sqrt 51.3%

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

      3. associate-*r* 51.3%

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

      4. fma-neg 51.4%

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

      5. add-sqr-sqrt 26.0%

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

      6. sqrt-unprod 42.7%

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

      7. sqr-neg 42.7%

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

      8. sqrt-unprod 18.1%

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

      9. add-sqr-sqrt 39.7%

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

   5. Applied egg-rr 39.7%

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

   6. Taylor expanded in x around inf 72.7%

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

      1. associate-*r* 72.1%

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

      2. *-commutative 72.1%

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

      3. associate-*r* 72.1%

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

   8. 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%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 7: 50.8% 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 99.5%

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

   1. associate-*l* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 51.6%

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

   1. mul-1-neg 51.6%

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

6. Simplified 51.6%

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

7. Final simplification 51.6%

   \[\leadsto -z \]

Alternative 8: 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 99.5%

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

   1. associate-*l* 99.4%

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

3. Simplified 99.4%

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

   1. associate-*r* 99.5%

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

   2. add-sqr-sqrt 52.5%

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

   3. associate-*r* 52.5%

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

   4. fma-neg 52.5%

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

   5. add-sqr-sqrt 24.8%

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

   6. sqrt-unprod 31.1%

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

   7. sqr-neg 31.1%

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

   8. sqrt-unprod 12.5%

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

   9. add-sqr-sqrt 26.7%

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

5. Applied egg-rr 26.7%

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

6. Taylor expanded in x around 0 2.3%

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

7. Final simplification 2.3%

   \[\leadsto z \]

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))