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

Percentage Accurate: 99.8% → 99.8%

Time: 5.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * 3.0d0) * y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * 3.0d0) * y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] * 3.0 + (-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. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. fmm-def 99.8%

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

7. Applied egg-rr 99.8%

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

Alternative 2: 69.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-141} \lor \neg \left(y \leq 7.4 \cdot 10^{-13}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.4e-141) (not (<= y 7.4e-13))) (* (* x y) 3.0) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.4e-141) || !(y <= 7.4e-13)) {
		tmp = (x * y) * 3.0;
	} 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) :: tmp
    if ((y <= (-4.4d-141)) .or. (.not. (y <= 7.4d-13))) then
        tmp = (x * y) * 3.0d0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.4e-141) || !(y <= 7.4e-13)) {
		tmp = (x * y) * 3.0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.4e-141) or not (y <= 7.4e-13):
		tmp = (x * y) * 3.0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.4e-141) || !(y <= 7.4e-13))
		tmp = Float64(Float64(x * y) * 3.0);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.4e-141) || ~((y <= 7.4e-13)))
		tmp = (x * y) * 3.0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.4e-141], N[Not[LessEqual[y, 7.4e-13]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] * 3.0), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if y < -4.40000000000000018e-141 or 7.39999999999999977e-13 < 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 x around inf 86.6%

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

      1. +-commutative 86.6%

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

      2. distribute-rgt-in 86.7%

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

      3. *-commutative 86.7%

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

      4. *-commutative 86.7%

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

      5. associate-*r* 86.7%

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

      6. metadata-eval 86.7%

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

      7. distribute-rgt-neg-in 86.7%

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

      8. *-commutative 86.7%

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

      9. *-commutative 86.7%

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

      10. associate-*r* 86.7%

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

      11. mul-1-neg 86.7%

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

      12. distribute-lft-neg-out 86.7%

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

      13. distribute-neg-in 86.7%

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

      14. distribute-rgt-in 86.6%

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

      15. distribute-rgt-neg-in 86.6%

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

      16. *-commutative 86.6%

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

      17. fma-define 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in x around inf 66.2%

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

   if -4.40000000000000018e-141 < y < 7.39999999999999977e-13

   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 82.6%

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

      1. mul-1-neg 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-141} \lor \neg \left(y \leq 7.4 \cdot 10^{-13}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-141}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \mathbf{elif}\;y \leq 5400000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.4e-141)
   (* (* x y) 3.0)
   (if (<= y 5400000.0) (- z) (* x (* y 3.0)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e-141) {
		tmp = (x * y) * 3.0;
	} else if (y <= 5400000.0) {
		tmp = -z;
	} else {
		tmp = x * (y * 3.0);
	}
	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) :: tmp
    if (y <= (-2.4d-141)) then
        tmp = (x * y) * 3.0d0
    else if (y <= 5400000.0d0) then
        tmp = -z
    else
        tmp = x * (y * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.4e-141) {
		tmp = (x * y) * 3.0;
	} else if (y <= 5400000.0) {
		tmp = -z;
	} else {
		tmp = x * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.4e-141:
		tmp = (x * y) * 3.0
	elif y <= 5400000.0:
		tmp = -z
	else:
		tmp = x * (y * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.4e-141)
		tmp = Float64(Float64(x * y) * 3.0);
	elseif (y <= 5400000.0)
		tmp = Float64(-z);
	else
		tmp = Float64(x * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.4e-141)
		tmp = (x * y) * 3.0;
	elseif (y <= 5400000.0)
		tmp = -z;
	else
		tmp = x * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.4e-141], N[(N[(x * y), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[y, 5400000.0], (-z), N[(x * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4000000000000001e-141

   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 inf 86.7%

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

      1. +-commutative 86.7%

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

      2. distribute-rgt-in 86.7%

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

      3. *-commutative 86.7%

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

      4. *-commutative 86.7%

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

      5. associate-*r* 86.8%

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

      6. metadata-eval 86.8%

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

      7. distribute-rgt-neg-in 86.8%

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

      8. *-commutative 86.8%

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

      9. *-commutative 86.8%

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

      10. associate-*r* 86.7%

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

      11. mul-1-neg 86.7%

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

      12. distribute-lft-neg-out 86.7%

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

      13. distribute-neg-in 86.7%

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

      14. distribute-rgt-in 86.7%

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

      15. distribute-rgt-neg-in 86.7%

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

      16. *-commutative 86.7%

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

      17. fma-define 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in x around inf 59.5%

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

   if -2.4000000000000001e-141 < y < 5.4e6

   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 81.6%

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

      1. mul-1-neg 81.6%

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

   7. Simplified 81.6%

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

   if 5.4e6 < 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 x around inf 86.4%

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

      1. +-commutative 86.4%

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

      2. distribute-rgt-in 86.4%

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

      3. *-commutative 86.4%

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

      4. *-commutative 86.4%

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

      5. associate-*r* 86.3%

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

      6. metadata-eval 86.3%

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

      7. distribute-rgt-neg-in 86.3%

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

      8. *-commutative 86.3%

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

      9. *-commutative 86.3%

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

      10. associate-*r* 86.4%

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

      11. mul-1-neg 86.4%

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

      12. distribute-lft-neg-out 86.4%

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

      13. distribute-neg-in 86.4%

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

      14. distribute-rgt-in 86.4%

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

      15. distribute-rgt-neg-in 86.4%

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

      16. *-commutative 86.4%

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

      17. fma-define 86.4%

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

   7. Simplified 86.4%

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

   8. Taylor expanded in y around inf 75.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-141}:\\ \;\;\;\;\left(x \cdot y\right) \cdot 3\\ \mathbf{elif}\;y \leq 5400000:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

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(y * Float64(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[(y * N[(x * 3.0), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y * 3.0), $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. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 7: 51.3% accurate, 3.5× 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 99.8%

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

Alternative 8: 2.2% accurate, 7.0× 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 99.8%

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

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

   5. add-sqr-sqrt 48.2%

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

   6. sqrt-unprod 58.7%

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

   7. sqr-neg 58.7%

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

   8. sqrt-unprod 23.6%

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

   9. add-sqr-sqrt 50.0%

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

6. Applied egg-rr 50.0%

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

7. Taylor expanded in y around 0 2.0%

   \[\leadsto \color{blue}{z} \]
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 2024191 
(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))