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

Percentage Accurate: 99.8% → 99.8%

Time: 6.4s

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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l* 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-neg 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 77.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := y \cdot \left(3 \cdot x\right)\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+52}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{elif}\;t_0 \leq -200000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 10^{-112}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (* 3.0 x))))
   (if (<= t_0 -4e+52)
     (* (* 3.0 y) x)
     (if (<= t_0 -200000000000.0)
       (- z)
       (if (<= t_0 -1e-37) t_0 (if (<= t_0 1e-112) (- z) (* 3.0 (* y x))))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = y * (3.0 * x);
	double tmp;
	if (t_0 <= -4e+52) {
		tmp = (3.0 * y) * x;
	} else if (t_0 <= -200000000000.0) {
		tmp = -z;
	} else if (t_0 <= -1e-37) {
		tmp = t_0;
	} else if (t_0 <= 1e-112) {
		tmp = -z;
	} else {
		tmp = 3.0 * (y * 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) :: t_0
    real(8) :: tmp
    t_0 = y * (3.0d0 * x)
    if (t_0 <= (-4d+52)) then
        tmp = (3.0d0 * y) * x
    else if (t_0 <= (-200000000000.0d0)) then
        tmp = -z
    else if (t_0 <= (-1d-37)) then
        tmp = t_0
    else if (t_0 <= 1d-112) then
        tmp = -z
    else
        tmp = 3.0d0 * (y * x)
    end if
    code = tmp
end function

Java

assert x < y && y < z;
public static double code(double x, double y, double z) {
	double t_0 = y * (3.0 * x);
	double tmp;
	if (t_0 <= -4e+52) {
		tmp = (3.0 * y) * x;
	} else if (t_0 <= -200000000000.0) {
		tmp = -z;
	} else if (t_0 <= -1e-37) {
		tmp = t_0;
	} else if (t_0 <= 1e-112) {
		tmp = -z;
	} else {
		tmp = 3.0 * (y * x);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = y * (3.0 * x)
	tmp = 0
	if t_0 <= -4e+52:
		tmp = (3.0 * y) * x
	elif t_0 <= -200000000000.0:
		tmp = -z
	elif t_0 <= -1e-37:
		tmp = t_0
	elif t_0 <= 1e-112:
		tmp = -z
	else:
		tmp = 3.0 * (y * x)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(y * Float64(3.0 * x))
	tmp = 0.0
	if (t_0 <= -4e+52)
		tmp = Float64(Float64(3.0 * y) * x);
	elseif (t_0 <= -200000000000.0)
		tmp = Float64(-z);
	elseif (t_0 <= -1e-37)
		tmp = t_0;
	elseif (t_0 <= 1e-112)
		tmp = Float64(-z);
	else
		tmp = Float64(3.0 * Float64(y * x));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = y * (3.0 * x);
	tmp = 0.0;
	if (t_0 <= -4e+52)
		tmp = (3.0 * y) * x;
	elseif (t_0 <= -200000000000.0)
		tmp = -z;
	elseif (t_0 <= -1e-37)
		tmp = t_0;
	elseif (t_0 <= 1e-112)
		tmp = -z;
	else
		tmp = 3.0 * (y * 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_] := Block[{t$95$0 = N[(y * N[(3.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+52], N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$0, -200000000000.0], (-z), If[LessEqual[t$95$0, -1e-37], t$95$0, If[LessEqual[t$95$0, 1e-112], (-z), N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 x 3) y) < -4e52

   1. Initial program 99.6%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{-1 \cdot z + 3 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-*r* 89.9%

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

      3. *-commutative 89.9%

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

   10. Simplified 89.9%

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

   if -4e52 < (*.f64 (*.f64 x 3) y) < -2e11 or -1.00000000000000007e-37 < (*.f64 (*.f64 x 3) y) < 9.9999999999999995e-113

   1. Initial program 100.0%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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 85.0%

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

      1. mul-1-neg 85.0%

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

   7. Simplified 85.0%

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

   if -2e11 < (*.f64 (*.f64 x 3) y) < -1.00000000000000007e-37

   1. Initial program 99.7%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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 99.7%

      \[\leadsto \color{blue}{-1 \cdot z + 3 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-def 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 75.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. unpow2 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*l* 0.0%

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

   10. Applied egg-rr 0.0%

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

       1. unpow2 0.0%

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

       2. add-sqr-sqrt 75.8%

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

       3. *-commutative 75.8%

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

       4. associate-*r* 75.6%

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

   12. Applied egg-rr 75.6%

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

   if 9.9999999999999995e-113 < (*.f64 (*.f64 x 3) y)

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot z + 3 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 81.2%

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(3 \cdot x\right) \leq -4 \cdot 10^{+52}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{elif}\;y \cdot \left(3 \cdot x\right) \leq -200000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \cdot \left(3 \cdot x\right) \leq -1 \cdot 10^{-37}:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{elif}\;y \cdot \left(3 \cdot x\right) \leq 10^{-112}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+82} \lor \neg \left(x \leq 1.96 \cdot 10^{-259}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5e+82) (not (<= x 1.96e-259))) (* 3.0 (* y x)) (- z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e+82) || !(x <= 1.96e-259)) {
		tmp = 3.0 * (y * x);
	} else {
		tmp = -z;
	}
	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 ((x <= (-5d+82)) .or. (.not. (x <= 1.96d-259))) then
        tmp = 3.0d0 * (y * x)
    else
        tmp = -z
    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 ((x <= -5e+82) || !(x <= 1.96e-259)) {
		tmp = 3.0 * (y * x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if (x <= -5e+82) or not (x <= 1.96e-259):
		tmp = 3.0 * (y * x)
	else:
		tmp = -z
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if ((x <= -5e+82) || !(x <= 1.96e-259))
		tmp = Float64(3.0 * Float64(y * x));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5e+82) || ~((x <= 1.96e-259)))
		tmp = 3.0 * (y * x);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := If[Or[LessEqual[x, -5e+82], N[Not[LessEqual[x, 1.96e-259]], $MachinePrecision]], N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000015e82 or 1.9600000000000001e-259 < x

   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. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot z + 3 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 66.3%

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

   if -5.00000000000000015e82 < x < 1.9600000000000001e-259

   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 74.1%

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

      1. mul-1-neg 74.1%

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

   7. Simplified 74.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+82} \lor \neg \left(x \leq 1.96 \cdot 10^{-259}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+82}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.96 \cdot 10^{-259}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= x -7.4e+82)
   (* (* 3.0 y) x)
   (if (<= x 1.96e-259) (- z) (* 3.0 (* y x)))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.4e+82) {
		tmp = (3.0 * y) * x;
	} else if (x <= 1.96e-259) {
		tmp = -z;
	} else {
		tmp = 3.0 * (y * 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 (x <= (-7.4d+82)) then
        tmp = (3.0d0 * y) * x
    else if (x <= 1.96d-259) then
        tmp = -z
    else
        tmp = 3.0d0 * (y * 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 (x <= -7.4e+82) {
		tmp = (3.0 * y) * x;
	} else if (x <= 1.96e-259) {
		tmp = -z;
	} else {
		tmp = 3.0 * (y * x);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	tmp = 0
	if x <= -7.4e+82:
		tmp = (3.0 * y) * x
	elif x <= 1.96e-259:
		tmp = -z
	else:
		tmp = 3.0 * (y * x)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	tmp = 0.0
	if (x <= -7.4e+82)
		tmp = Float64(Float64(3.0 * y) * x);
	elseif (x <= 1.96e-259)
		tmp = Float64(-z);
	else
		tmp = Float64(3.0 * Float64(y * 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 (x <= -7.4e+82)
		tmp = (3.0 * y) * x;
	elseif (x <= 1.96e-259)
		tmp = -z;
	else
		tmp = 3.0 * (y * 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[LessEqual[x, -7.4e+82], N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.96e-259], (-z), N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.4000000000000005e82

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot z + 3 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*r* 81.3%

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

      3. *-commutative 81.3%

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

   10. Simplified 81.3%

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

   if -7.4000000000000005e82 < x < 1.9600000000000001e-259

   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 74.1%

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

      1. mul-1-neg 74.1%

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

   7. Simplified 74.1%

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

   if 1.9600000000000001e-259 < x

   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. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot z + 3 \cdot \left(x \cdot y\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 59.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+82}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{elif}\;x \leq 1.96 \cdot 10^{-259}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp = code(x, y, z)
	tmp = (3.0 * (y * x)) - 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[(y * x), $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* 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, and z 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 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
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. Final simplification 99.8%

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

Alternative 7: 49.7% 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 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
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. Taylor expanded in x around 0 51.0%

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

   1. mul-1-neg 51.0%

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

7. Simplified 51.0%

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

8. Final simplification 51.0%

   \[\leadsto -z \]
9. 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 2024019 
(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))