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

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

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, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (fma (* y 3.0) x (- z)))

C

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

Julia

function code(x, y, z)
	return fma(Float64(y * 3.0), x, Float64(-z))
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 3.0), $MachinePrecision] * x + (-z)), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 78.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+75} \lor \neg \left(t\_0 \leq 10^{-14}\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x 3.0) y)))
   (if (or (<= t_0 -2e+75) (not (<= t_0 1e-14))) (* (* 3.0 y) x) (- z))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if ((t_0 <= -2e+75) || !(t_0 <= 1e-14)) {
		tmp = (3.0 * y) * x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

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 = (x * 3.0d0) * y
    if ((t_0 <= (-2d+75)) .or. (.not. (t_0 <= 1d-14))) then
        tmp = (3.0d0 * y) * x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if ((t_0 <= -2e+75) || !(t_0 <= 1e-14)) {
		tmp = (3.0 * y) * x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 3.0) * y
	tmp = 0
	if (t_0 <= -2e+75) or not (t_0 <= 1e-14):
		tmp = (3.0 * y) * x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	tmp = 0.0
	if ((t_0 <= -2e+75) || !(t_0 <= 1e-14))
		tmp = Float64(Float64(3.0 * y) * x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if ((t_0 <= -2e+75) || ~((t_0 <= 1e-14)))
		tmp = (3.0 * y) * x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+75], N[Not[LessEqual[t$95$0, 1e-14]], $MachinePrecision]], N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -1.99999999999999985e75 or 9.99999999999999999e-15 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 32.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\left(9 \cdot \left(x \cdot x\right)\right) \cdot y}{\mathsf{fma}\left(y, 3 \cdot x, z\right)}, -z \cdot \frac{z}{\mathsf{fma}\left(y, 3 \cdot x, z\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\left(9 \cdot \left(x \cdot x\right)\right) \cdot y}{\mathsf{fma}\left(y, 3 \cdot x, z\right)}, -z \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 31.8%

      \[\leadsto \mathsf{fma}\left(y, \frac{\left(9 \cdot \left(x \cdot x\right)\right) \cdot y}{\mathsf{fma}\left(y, 3 \cdot x, z\right)}, -z \cdot \color{blue}{1}\right) \]

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 85.4

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

   10. Applied rewrites 85.4%

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

   12. Applied rewrites 85.5%

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

   if -1.99999999999999985e75 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 9.99999999999999999e-15

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -2 \cdot 10^{+75} \lor \neg \left(\left(x \cdot 3\right) \cdot y \leq 10^{-14}\right):\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;\left(3 \cdot x\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 10^{-14}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x 3.0) y)))
   (if (<= t_0 -2e+75)
     (* (* 3.0 x) y)
     (if (<= t_0 1e-14) (- z) (* (* 3.0 y) x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -2e+75) {
		tmp = (3.0 * x) * y;
	} else if (t_0 <= 1e-14) {
		tmp = -z;
	} else {
		tmp = (3.0 * y) * x;
	}
	return tmp;
}

Fortran

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 = (x * 3.0d0) * y
    if (t_0 <= (-2d+75)) then
        tmp = (3.0d0 * x) * y
    else if (t_0 <= 1d-14) then
        tmp = -z
    else
        tmp = (3.0d0 * y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -2e+75) {
		tmp = (3.0 * x) * y;
	} else if (t_0 <= 1e-14) {
		tmp = -z;
	} else {
		tmp = (3.0 * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 3.0) * y
	tmp = 0
	if t_0 <= -2e+75:
		tmp = (3.0 * x) * y
	elif t_0 <= 1e-14:
		tmp = -z
	else:
		tmp = (3.0 * y) * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	tmp = 0.0
	if (t_0 <= -2e+75)
		tmp = Float64(Float64(3.0 * x) * y);
	elseif (t_0 <= 1e-14)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(3.0 * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if (t_0 <= -2e+75)
		tmp = (3.0 * x) * y;
	elseif (t_0 <= 1e-14)
		tmp = -z;
	else
		tmp = (3.0 * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+75], N[(N[(3.0 * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 1e-14], (-z), N[(N[(3.0 * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -1.99999999999999985e75

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-neg.f64 N/A

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

      7. neg-sub0 N/A

         \[\leadsto \left(3 \cdot x\right) \cdot y + \color{blue}{\left(0 - z\right)} \]

      8. flip3-- N/A

         \[\leadsto \left(3 \cdot x\right) \cdot y + \color{blue}{\frac{{0}^{3} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}} \]

      9. metadata-eval N/A

         \[\leadsto \left(3 \cdot x\right) \cdot y + \frac{\color{blue}{0} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)} \]

      10. neg-sub0 N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)} \]

      11. distribute-neg-frac N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{{z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right)} \]

      12. sqr-pow N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      13. pow-prod-down N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(z \cdot z\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      14. sqr-neg N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      15. lift-neg.f64 N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{{\left(\color{blue}{\left(-z\right)} \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      16. lift-neg.f64 N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{{\left(\left(-z\right) \cdot \color{blue}{\left(-z\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      17. pow-prod-down N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(-z\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(-z\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      18. sqr-pow N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{\color{blue}{{\left(-z\right)}^{3}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      19. lift-neg.f64 N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      20. cube-neg N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{neg}\left({z}^{3}\right)}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      21. neg-sub0 N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{\color{blue}{0 - {z}^{3}}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      22. metadata-eval N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\frac{\color{blue}{{0}^{3}} - {z}^{3}}{0 \cdot 0 + \left(z \cdot z + 0 \cdot z\right)}\right)\right) \]

      23. flip3-- N/A

          \[\leadsto \left(3 \cdot x\right) \cdot y + \left(\mathsf{neg}\left(\color{blue}{\left(0 - z\right)}\right)\right) \]

      24. neg-sub0 N/A

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

   6. Applied rewrites 90.4%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 90.3

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

   9. Applied rewrites 90.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 90.8%

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

   if -1.99999999999999985e75 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 9.99999999999999999e-15

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if 9.99999999999999999e-15 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 98.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\left(9 \cdot \left(x \cdot x\right)\right) \cdot y}{\mathsf{fma}\left(y, 3 \cdot x, z\right)}, -z \cdot \frac{z}{\mathsf{fma}\left(y, 3 \cdot x, z\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\left(9 \cdot \left(x \cdot x\right)\right) \cdot y}{\mathsf{fma}\left(y, 3 \cdot x, z\right)}, -z \cdot \color{blue}{1}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 35.1%

      \[\leadsto \mathsf{fma}\left(y, \frac{\left(9 \cdot \left(x \cdot x\right)\right) \cdot y}{\mathsf{fma}\left(y, 3 \cdot x, z\right)}, -z \cdot \color{blue}{1}\right) \]

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 81.5

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

   10. Applied rewrites 81.5%

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

   12. Applied rewrites 81.6%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z) :precision binary64 (- (* (* y x) 3.0) z))

C

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

Java

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

Python

def code(x, y, z):
	return ((y * x) * 3.0) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y * x) * 3.0) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y * x) * 3.0) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y * x), $MachinePrecision] * 3.0), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 5: 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]

Derivation

1. Initial program 99.4%

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

Alternative 6: 50.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

double code(double x, double y, double z) {
	return -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 = -z
end function

Java

public static double code(double x, double y, double z) {
	return -z;
}

Python

def code(x, y, z):
	return -z

Julia

function code(x, y, z)
	return Float64(-z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -z;
end

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 45.3

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

5. Applied rewrites 45.3%

   \[\leadsto \color{blue}{-z} \]
6. Add Preprocessing

Alternative 7: 2.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 z)

C

double code(double x, double y, double z) {
	return 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 = z
end function

Java

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

Python

def code(x, y, z):
	return z

Julia

function code(x, y, z)
	return z
end

MATLAB

function tmp = code(x, y, z)
	tmp = z;
end

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 45.3

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

5. Applied rewrites 45.3%

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

7. Applied rewrites 2.5%

   \[\leadsto z \]
8. Add Preprocessing

Developer Target 1: 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 2024313 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* x (* 3 y)) z))

  (- (* (* x 3.0) y) z))