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

Percentage Accurate: 99.7% → 99.7%

Time: 4.0s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot 27\right) \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* x 27.0) y))

C

double code(double x, double y) {
	return (x * 27.0) * y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * 27.0d0) * y
end function

Java

public static double code(double x, double y) {
	return (x * 27.0) * y;
}

Python

def code(x, y):
	return (x * 27.0) * y

Julia

function code(x, y)
	return Float64(Float64(x * 27.0) * y)
end

MATLAB

function tmp = code(x, y)
	tmp = (x * 27.0) * y;
end

Wolfram

code[x_, y_] := N[(N[(x * 27.0), $MachinePrecision] * y), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 27\right) \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (* x 27.0) y))

C

double code(double x, double y) {
	return (x * 27.0) * y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * 27.0d0) * y
end function

Java

public static double code(double x, double y) {
	return (x * 27.0) * y;
}

Python

def code(x, y):
	return (x * 27.0) * y

Julia

function code(x, y)
	return Float64(Float64(x * 27.0) * y)
end

MATLAB

function tmp = code(x, y)
	tmp = (x * 27.0) * y;
end

Wolfram

code[x_, y_] := N[(N[(x * 27.0), $MachinePrecision] * y), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ x\_s \cdot \left(y\_s \cdot \left(\left(x\_m \cdot 27\right) \cdot y\_m\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m)
 :precision binary64
 (* x_s (* y_s (* (* x_m 27.0) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m);
double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * ((x_m * 27.0) * y_m));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = x_s * (y_s * ((x_m * 27.0d0) * y_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m;
public static double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * ((x_m * 27.0) * y_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m] = sort([x_m, y_m])
def code(x_s, y_s, x_m, y_m):
	return x_s * (y_s * ((x_m * 27.0) * y_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m = sort([x_m, y_m])
function code(x_s, y_s, x_m, y_m)
	return Float64(x_s * Float64(y_s * Float64(Float64(x_m * 27.0) * y_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m = num2cell(sort([x_m, y_m])){:}
function tmp = code(x_s, y_s, x_m, y_m)
	tmp = x_s * (y_s * ((x_m * 27.0) * y_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_] := N[(x$95$s * N[(y$95$s * N[(N[(x$95$m * 27.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\left(x \cdot 27\right) \cdot y \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 17.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(x\_m \cdot 27\right) \cdot y\_m \leq 4 \cdot 10^{-223}:\\ \;\;\;\;x\_m \cdot \left(-y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;27 \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m)
 :precision binary64
 (*
  x_s
  (* y_s (if (<= (* (* x_m 27.0) y_m) 4e-223) (* x_m (- y_m)) (* 27.0 y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m);
double code(double x_s, double y_s, double x_m, double y_m) {
	double tmp;
	if (((x_m * 27.0) * y_m) <= 4e-223) {
		tmp = x_m * -y_m;
	} else {
		tmp = 27.0 * y_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((x_m * 27.0d0) * y_m) <= 4d-223) then
        tmp = x_m * -y_m
    else
        tmp = 27.0d0 * y_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m;
public static double code(double x_s, double y_s, double x_m, double y_m) {
	double tmp;
	if (((x_m * 27.0) * y_m) <= 4e-223) {
		tmp = x_m * -y_m;
	} else {
		tmp = 27.0 * y_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m] = sort([x_m, y_m])
def code(x_s, y_s, x_m, y_m):
	tmp = 0
	if ((x_m * 27.0) * y_m) <= 4e-223:
		tmp = x_m * -y_m
	else:
		tmp = 27.0 * y_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m = sort([x_m, y_m])
function code(x_s, y_s, x_m, y_m)
	tmp = 0.0
	if (Float64(Float64(x_m * 27.0) * y_m) <= 4e-223)
		tmp = Float64(x_m * Float64(-y_m));
	else
		tmp = Float64(27.0 * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m = num2cell(sort([x_m, y_m])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m)
	tmp = 0.0;
	if (((x_m * 27.0) * y_m) <= 4e-223)
		tmp = x_m * -y_m;
	else
		tmp = 27.0 * y_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(x$95$m * 27.0), $MachinePrecision] * y$95$m), $MachinePrecision], 4e-223], N[(x$95$m * (-y$95$m)), $MachinePrecision], N[(27.0 * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 27 binary64)) y) < 3.9999999999999999e-223

   1. Initial program 99.2%

      \[\left(x \cdot 27\right) \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 21.4

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

   5. Simplified 21.4%

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

   if 3.9999999999999999e-223 < (*.f64 (*.f64 x #s(literal 27 binary64)) y)

   1. Initial program 98.8%

      \[\left(x \cdot 27\right) \cdot y \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{x \cdot \left(27 \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(27 \cdot y\right) \cdot x} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(27 \cdot y\right) \cdot x} \]

      4. lower-*.f64 99.8

         \[\leadsto \color{blue}{\left(27 \cdot y\right)} \cdot x \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\left(27 \cdot y\right) \cdot x} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{27 \cdot y} \]
   6. Step-by-step derivation

      1. lower-*.f64 3.6

         \[\leadsto \color{blue}{27 \cdot y} \]

   7. Simplified 3.6%

      \[\leadsto \color{blue}{27 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 27\right) \cdot y \leq 4 \cdot 10^{-223}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;27 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ x\_s \cdot \left(y\_s \cdot \left(x\_m \cdot \left(27 \cdot y\_m\right)\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m)
 :precision binary64
 (* x_s (* y_s (* x_m (* 27.0 y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m);
double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * (x_m * (27.0 * y_m)));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = x_s * (y_s * (x_m * (27.0d0 * y_m)))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m;
public static double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * (x_m * (27.0 * y_m)));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m] = sort([x_m, y_m])
def code(x_s, y_s, x_m, y_m):
	return x_s * (y_s * (x_m * (27.0 * y_m)))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m = sort([x_m, y_m])
function code(x_s, y_s, x_m, y_m)
	return Float64(x_s * Float64(y_s * Float64(x_m * Float64(27.0 * y_m))))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m = num2cell(sort([x_m, y_m])){:}
function tmp = code(x_s, y_s, x_m, y_m)
	tmp = x_s * (y_s * (x_m * (27.0 * y_m)));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_] := N[(x$95$s * N[(y$95$s * N[(x$95$m * N[(27.0 * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\left(x \cdot 27\right) \cdot y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(27 \cdot y\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(27 \cdot y\right) \cdot x} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(27 \cdot y\right) \cdot x} \]

   4. lower-*.f64 99.7

      \[\leadsto \color{blue}{\left(27 \cdot y\right)} \cdot x \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(27 \cdot y\right) \cdot x} \]

5. Final simplification 99.7%

   \[\leadsto x \cdot \left(27 \cdot y\right) \]
6. Add Preprocessing

Alternative 4: 6.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ x\_s \cdot \left(y\_s \cdot \left(27 \cdot y\_m\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m) :precision binary64 (* x_s (* y_s (* 27.0 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m);
double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * (27.0 * y_m));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = x_s * (y_s * (27.0d0 * y_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m;
public static double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * (27.0 * y_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m] = sort([x_m, y_m])
def code(x_s, y_s, x_m, y_m):
	return x_s * (y_s * (27.0 * y_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m = sort([x_m, y_m])
function code(x_s, y_s, x_m, y_m)
	return Float64(x_s * Float64(y_s * Float64(27.0 * y_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m = num2cell(sort([x_m, y_m])){:}
function tmp = code(x_s, y_s, x_m, y_m)
	tmp = x_s * (y_s * (27.0 * y_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_] := N[(x$95$s * N[(y$95$s * N[(27.0 * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\left(x \cdot 27\right) \cdot y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(27 \cdot y\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(27 \cdot y\right) \cdot x} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(27 \cdot y\right) \cdot x} \]

   4. lower-*.f64 99.7

      \[\leadsto \color{blue}{\left(27 \cdot y\right)} \cdot x \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(27 \cdot y\right) \cdot x} \]

5. Taylor expanded in y around inf

   \[\leadsto \color{blue}{27 \cdot y} \]
6. Step-by-step derivation

   1. lower-*.f64 4.1

      \[\leadsto \color{blue}{27 \cdot y} \]

7. Simplified 4.1%

   \[\leadsto \color{blue}{27 \cdot y} \]
8. Add Preprocessing

Alternative 5: 5.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ x\_s \cdot \left(y\_s \cdot \left(x\_m \cdot 27\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m) :precision binary64 (* x_s (* y_s (* x_m 27.0))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m);
double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * (x_m * 27.0));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = x_s * (y_s * (x_m * 27.0d0))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m;
public static double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * (x_m * 27.0));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m] = sort([x_m, y_m])
def code(x_s, y_s, x_m, y_m):
	return x_s * (y_s * (x_m * 27.0))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m = sort([x_m, y_m])
function code(x_s, y_s, x_m, y_m)
	return Float64(x_s * Float64(y_s * Float64(x_m * 27.0)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m = num2cell(sort([x_m, y_m])){:}
function tmp = code(x_s, y_s, x_m, y_m)
	tmp = x_s * (y_s * (x_m * 27.0));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_] := N[(x$95$s * N[(y$95$s * N[(x$95$m * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\left(x \cdot 27\right) \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{27 \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 4.5

      \[\leadsto \color{blue}{27 \cdot x} \]

5. Simplified 4.5%

   \[\leadsto \color{blue}{27 \cdot x} \]

6. Final simplification 4.5%

   \[\leadsto x \cdot 27 \]
7. Add Preprocessing

Alternative 6: 2.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m] = \mathsf{sort}([x_m, y_m])\\ \\ x\_s \cdot \left(y\_s \cdot \left(-x\_m\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m) :precision binary64 (* x_s (* y_s (- x_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m);
double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * -x_m);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    code = x_s * (y_s * -x_m)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m;
public static double code(double x_s, double y_s, double x_m, double y_m) {
	return x_s * (y_s * -x_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m] = sort([x_m, y_m])
def code(x_s, y_s, x_m, y_m):
	return x_s * (y_s * -x_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m = sort([x_m, y_m])
function code(x_s, y_s, x_m, y_m)
	return Float64(x_s * Float64(y_s * Float64(-x_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m = num2cell(sort([x_m, y_m])){:}
function tmp = code(x_s, y_s, x_m, y_m)
	tmp = x_s * (y_s * -x_m);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m and y_m should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_] := N[(x$95$s * N[(y$95$s * (-x$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

   \[\left(x \cdot 27\right) \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{27 \cdot x} \]
4. Step-by-step derivation

   1. lower-*.f64 4.5

      \[\leadsto \color{blue}{27 \cdot x} \]

5. Simplified 4.5%

   \[\leadsto \color{blue}{27 \cdot x} \]

6. Taylor expanded in x around -inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 4.2

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

8. Simplified 4.2%

   \[\leadsto \color{blue}{-x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x y)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, F"
  :precision binary64
  (* (* x 27.0) y))