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

Percentage Accurate: 99.8% → 99.8%

Time: 5.5s

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: 71.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-59} \lor \neg \left(y \leq 2.8 \cdot 10^{-69} \lor \neg \left(y \leq 4.5 \cdot 10^{-48}\right) \land y \leq 2.4 \cdot 10^{-23}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.3e-59)
         (not (or (<= y 2.8e-69) (and (not (<= y 4.5e-48)) (<= y 2.4e-23)))))
   (* 3.0 (* y x))
   (- z)))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.3e-59) || !((y <= 2.8e-69) || (!(y <= 4.5e-48) && (y <= 2.4e-23)))) {
		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 ((y <= (-2.3d-59)) .or. (.not. (y <= 2.8d-69) .or. (.not. (y <= 4.5d-48)) .and. (y <= 2.4d-23))) 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 ((y <= -2.3e-59) || !((y <= 2.8e-69) || (!(y <= 4.5e-48) && (y <= 2.4e-23)))) {
		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 (y <= -2.3e-59) or not ((y <= 2.8e-69) or (not (y <= 4.5e-48) and (y <= 2.4e-23))):
		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 ((y <= -2.3e-59) || !((y <= 2.8e-69) || (!(y <= 4.5e-48) && (y <= 2.4e-23))))
		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 ((y <= -2.3e-59) || ~(((y <= 2.8e-69) || (~((y <= 4.5e-48)) && (y <= 2.4e-23)))))
		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[y, -2.3e-59], N[Not[Or[LessEqual[y, 2.8e-69], And[N[Not[LessEqual[y, 4.5e-48]], $MachinePrecision], LessEqual[y, 2.4e-23]]]], $MachinePrecision]], N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if y < -2.29999999999999979e-59 or 2.79999999999999979e-69 < y < 4.49999999999999988e-48 or 2.39999999999999996e-23 < y

   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 y around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-define 99.7%

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

      3. mul-1-neg 99.7%

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

      4. fma-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 72.2%

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

   if -2.29999999999999979e-59 < y < 2.79999999999999979e-69 or 4.49999999999999988e-48 < y < 2.39999999999999996e-23

   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 79.9%

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

      1. mul-1-neg 79.9%

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

   7. Simplified 79.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{-59} \lor \neg \left(y \leq 2.8 \cdot 10^{-69} \lor \neg \left(y \leq 4.5 \cdot 10^{-48}\right) \land y \leq 2.4 \cdot 10^{-23}\right):\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := 3 \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{-59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-69}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

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 (* 3.0 (* y x))))
   (if (<= y -2.7e-59)
     t_0
     (if (<= y 2.65e-69)
       (- z)
       (if (<= y 5.5e-48) t_0 (if (<= y 4.5e-23) (- z) (* (* 3.0 y) x)))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = 3.0 * (y * x);
	double tmp;
	if (y <= -2.7e-59) {
		tmp = t_0;
	} else if (y <= 2.65e-69) {
		tmp = -z;
	} else if (y <= 5.5e-48) {
		tmp = t_0;
	} else if (y <= 4.5e-23) {
		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 = 3.0d0 * (y * x)
    if (y <= (-2.7d-59)) then
        tmp = t_0
    else if (y <= 2.65d-69) then
        tmp = -z
    else if (y <= 5.5d-48) then
        tmp = t_0
    else if (y <= 4.5d-23) 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 = 3.0 * (y * x);
	double tmp;
	if (y <= -2.7e-59) {
		tmp = t_0;
	} else if (y <= 2.65e-69) {
		tmp = -z;
	} else if (y <= 5.5e-48) {
		tmp = t_0;
	} else if (y <= 4.5e-23) {
		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 = 3.0 * (y * x)
	tmp = 0
	if y <= -2.7e-59:
		tmp = t_0
	elif y <= 2.65e-69:
		tmp = -z
	elif y <= 5.5e-48:
		tmp = t_0
	elif y <= 4.5e-23:
		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(3.0 * Float64(y * x))
	tmp = 0.0
	if (y <= -2.7e-59)
		tmp = t_0;
	elseif (y <= 2.65e-69)
		tmp = Float64(-z);
	elseif (y <= 5.5e-48)
		tmp = t_0;
	elseif (y <= 4.5e-23)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(3.0 * y) * x);
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = 3.0 * (y * x);
	tmp = 0.0;
	if (y <= -2.7e-59)
		tmp = t_0;
	elseif (y <= 2.65e-69)
		tmp = -z;
	elseif (y <= 5.5e-48)
		tmp = t_0;
	elseif (y <= 4.5e-23)
		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[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.7e-59], t$95$0, If[LessEqual[y, 2.65e-69], (-z), If[LessEqual[y, 5.5e-48], t$95$0, If[LessEqual[y, 4.5e-23], (-z), N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.6999999999999999e-59 or 2.65e-69 < y < 5.50000000000000047e-48

   1. Initial program 99.8%

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

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

      1. +-commutative 99.7%

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

      2. fma-define 99.7%

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

      3. mul-1-neg 99.7%

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

      4. fma-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 69.7%

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

   if -2.6999999999999999e-59 < y < 2.65e-69 or 5.50000000000000047e-48 < y < 4.49999999999999975e-23

   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 79.9%

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

      1. mul-1-neg 79.9%

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

   7. Simplified 79.9%

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

   if 4.49999999999999975e-23 < y

   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 y around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-define 99.7%

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

      3. mul-1-neg 99.7%

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

      4. fma-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 75.3%

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

      1. associate-*r* 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*r* 75.3%

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

   10. Simplified 75.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-59}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{-69}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-48}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-48}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (if (<= y -6.5e-58)
   (* y (* 3.0 x))
   (if (<= y 2.2e-69)
     (- z)
     (if (<= y 5e-48)
       (* 3.0 (* y x))
       (if (<= y 4.4e-23) (- z) (* (* 3.0 y) x))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double tmp;
	if (y <= -6.5e-58) {
		tmp = y * (3.0 * x);
	} else if (y <= 2.2e-69) {
		tmp = -z;
	} else if (y <= 5e-48) {
		tmp = 3.0 * (y * x);
	} else if (y <= 4.4e-23) {
		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 (y <= (-6.5d-58)) then
        tmp = y * (3.0d0 * x)
    else if (y <= 2.2d-69) then
        tmp = -z
    else if (y <= 5d-48) then
        tmp = 3.0d0 * (y * x)
    else if (y <= 4.4d-23) 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 (y <= -6.5e-58) {
		tmp = y * (3.0 * x);
	} else if (y <= 2.2e-69) {
		tmp = -z;
	} else if (y <= 5e-48) {
		tmp = 3.0 * (y * x);
	} else if (y <= 4.4e-23) {
		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 y <= -6.5e-58:
		tmp = y * (3.0 * x)
	elif y <= 2.2e-69:
		tmp = -z
	elif y <= 5e-48:
		tmp = 3.0 * (y * x)
	elif y <= 4.4e-23:
		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 (y <= -6.5e-58)
		tmp = Float64(y * Float64(3.0 * x));
	elseif (y <= 2.2e-69)
		tmp = Float64(-z);
	elseif (y <= 5e-48)
		tmp = Float64(3.0 * Float64(y * x));
	elseif (y <= 4.4e-23)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(3.0 * 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 (y <= -6.5e-58)
		tmp = y * (3.0 * x);
	elseif (y <= 2.2e-69)
		tmp = -z;
	elseif (y <= 5e-48)
		tmp = 3.0 * (y * x);
	elseif (y <= 4.4e-23)
		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[y, -6.5e-58], N[(y * N[(3.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-69], (-z), If[LessEqual[y, 5e-48], N[(3.0 * N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e-23], (-z), N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.49999999999999964e-58

   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 y around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.7%

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

      3. mul-1-neg 99.7%

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

      4. fma-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in y around inf 69.4%

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

      1. associate-*r* 69.5%

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

   10. Simplified 69.5%

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

   if -6.49999999999999964e-58 < y < 2.2e-69 or 4.9999999999999999e-48 < y < 4.3999999999999999e-23

   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 79.9%

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

      1. mul-1-neg 79.9%

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

   7. Simplified 79.9%

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

   if 2.2e-69 < y < 4.9999999999999999e-48

   1. Initial program 99.2%

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

      1. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-define 99.2%

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

      3. mul-1-neg 99.2%

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

      4. fma-neg 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in y around inf 75.7%

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

   if 4.3999999999999999e-23 < y

   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 y around inf 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-define 99.7%

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

      3. mul-1-neg 99.7%

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

      4. fma-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 75.3%

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

      1. associate-*r* 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*r* 75.3%

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

   10. Simplified 75.3%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-58}:\\ \;\;\;\;y \cdot \left(3 \cdot x\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-48}:\\ \;\;\;\;3 \cdot \left(y \cdot x\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \]
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: 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 99.8%

   \[\left(x \cdot 3\right) \cdot y - z \]
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 50.5%

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

   1. mul-1-neg 50.5%

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

7. Simplified 50.5%

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

8. Final simplification 50.5%

   \[\leadsto -z \]
9. Add Preprocessing

Alternative 7: 2.3% accurate, 7.0× 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 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. Step-by-step derivation

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*r* 99.8%

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

   4. fma-neg 99.8%

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

   5. add-sqr-sqrt 51.6%

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

   6. sqrt-unprod 61.2%

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

   7. sqr-neg 61.2%

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

   8. sqrt-unprod 24.2%

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

   9. add-sqr-sqrt 49.1%

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

6. Applied egg-rr 49.1%

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

7. Taylor expanded in y around 0 2.3%

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

8. Final simplification 2.3%

   \[\leadsto z \]
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 2024058 
(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))