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

Percentage Accurate: 99.8% → 99.8%

Time: 5.8s

Alternatives: 6

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

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

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 99.9%

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

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

Alternative 2: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 10^{+73}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, and z should be sorted in increasing order before calling this function.
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* x 3.0) y)))
   (if (<= t_0 -1e-73) t_0 (if (<= t_0 1e+73) (- z) (* x (* 3.0 y))))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -1e-73) {
		tmp = t_0;
	} else if (t_0 <= 1e+73) {
		tmp = -z;
	} else {
		tmp = x * (3.0 * y);
	}
	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 = (x * 3.0d0) * y
    if (t_0 <= (-1d-73)) then
        tmp = t_0
    else if (t_0 <= 1d+73) then
        tmp = -z
    else
        tmp = x * (3.0d0 * y)
    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 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -1e-73) {
		tmp = t_0;
	} else if (t_0 <= 1e+73) {
		tmp = -z;
	} else {
		tmp = x * (3.0 * y);
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (x * 3.0) * y
	tmp = 0
	if t_0 <= -1e-73:
		tmp = t_0
	elif t_0 <= 1e+73:
		tmp = -z
	else:
		tmp = x * (3.0 * y)
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	tmp = 0.0
	if (t_0 <= -1e-73)
		tmp = t_0;
	elseif (t_0 <= 1e+73)
		tmp = Float64(-z);
	else
		tmp = Float64(x * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if (t_0 <= -1e-73)
		tmp = t_0;
	elseif (t_0 <= 1e+73)
		tmp = -z;
	else
		tmp = x * (3.0 * y);
	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[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-73], t$95$0, If[LessEqual[t$95$0, 1e+73], (-z), N[(x * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -9.99999999999999997e-74

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}}, 3, -z\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}}, 3, -z\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\left(y \cdot x\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}, 3, -z\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y \cdot x}} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}, 3, -z\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y \cdot x}} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}, 3, -z\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{y \cdot x} \cdot {\left(y \cdot x\right)}^{\color{blue}{\frac{1}{2}}}, 3, -z\right) \]

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]
   8. 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 78.9

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

   9. Applied rewrites 78.9%

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

   11. Applied rewrites 79.2%

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

   if -9.99999999999999997e-74 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 9.99999999999999983e72

   1. Initial program 100.0%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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 84.2

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

   5. Applied rewrites 84.2%

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

   if 9.99999999999999983e72 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}}, 3, -z\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}}, 3, -z\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\left(y \cdot x\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}, 3, -z\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y \cdot x}} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}, 3, -z\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y \cdot x}} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}, 3, -z\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{y \cdot x} \cdot {\left(y \cdot x\right)}^{\color{blue}{\frac{1}{2}}}, 3, -z\right) \]

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 99.5

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

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]
   8. 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 91.8

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

   9. Applied rewrites 91.8%

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

   11. Applied rewrites 91.8%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \mathbf{elif}\;\left(x \cdot 3\right) \cdot y \leq 10^{+73}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} [x, y, z] = \mathsf{sort}([x, y, z])\\ \\ \begin{array}{l} t_0 := \left(x \cdot 3\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 10^{+73}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* (* x 3.0) y)))
   (if (<= t_0 -1e-73) t_0 (if (<= t_0 1e+73) (- z) t_0))))

C

assert(x < y && y < z);
double code(double x, double y, double z) {
	double t_0 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -1e-73) {
		tmp = t_0;
	} else if (t_0 <= 1e+73) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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 = (x * 3.0d0) * y
    if (t_0 <= (-1d-73)) then
        tmp = t_0
    else if (t_0 <= 1d+73) then
        tmp = -z
    else
        tmp = t_0
    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 = (x * 3.0) * y;
	double tmp;
	if (t_0 <= -1e-73) {
		tmp = t_0;
	} else if (t_0 <= 1e+73) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y, z] = sort([x, y, z])
def code(x, y, z):
	t_0 = (x * 3.0) * y
	tmp = 0
	if t_0 <= -1e-73:
		tmp = t_0
	elif t_0 <= 1e+73:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

x, y, z = sort([x, y, z])
function code(x, y, z)
	t_0 = Float64(Float64(x * 3.0) * y)
	tmp = 0.0
	if (t_0 <= -1e-73)
		tmp = t_0;
	elseif (t_0 <= 1e+73)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y, z = num2cell(sort([x, y, z])){:}
function tmp_2 = code(x, y, z)
	t_0 = (x * 3.0) * y;
	tmp = 0.0;
	if (t_0 <= -1e-73)
		tmp = t_0;
	elseif (t_0 <= 1e+73)
		tmp = -z;
	else
		tmp = t_0;
	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[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-73], t$95$0, If[LessEqual[t$95$0, 1e+73], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 x #s(literal 3 binary64)) y) < -9.99999999999999997e-74 or 9.99999999999999983e72 < (*.f64 (*.f64 x #s(literal 3 binary64)) y)

   1. Initial program 99.9%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}}, 3, -z\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}}, 3, -z\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({\left(y \cdot x\right)}^{\color{blue}{\frac{1}{2}}} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}, 3, -z\right) \]

      5. unpow1/2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y \cdot x}} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}, 3, -z\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{y \cdot x}} \cdot {\left(y \cdot x\right)}^{\left(\frac{1}{2}\right)}, 3, -z\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{y \cdot x} \cdot {\left(y \cdot x\right)}^{\color{blue}{\frac{1}{2}}}, 3, -z\right) \]

      8. unpow1/2 N/A

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

      9. lower-sqrt.f64 37.0

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

   6. Applied rewrites 37.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \color{blue}{3 \cdot \left(x \cdot y\right)} \]
   8. 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 83.7

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

   9. Applied rewrites 83.7%

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

   11. Applied rewrites 83.9%

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

   if -9.99999999999999997e-74 < (*.f64 (*.f64 x #s(literal 3 binary64)) y) < 9.99999999999999983e72

   1. Initial program 100.0%

      \[\left(x \cdot 3\right) \cdot y - z \]
   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 84.2

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

   5. Applied rewrites 84.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot 3\right) \cdot y \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \mathbf{elif}\;\left(x \cdot 3\right) \cdot y \leq 10^{+73}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 3\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x, y, and z should be sorted in increasing order before calling this function.
code[x_, y_, z_] := N[(N[(N[(x * 3.0), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

Alternative 5: 51.9% accurate, 4.7× 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 99.9%

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

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

5. Applied rewrites 50.5%

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

Alternative 6: 2.2% accurate, 14.0× 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 99.9%

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

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

5. Applied rewrites 50.5%

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

7. Applied rewrites 2.3%

   \[\leadsto \color{blue}{z} \]
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 2024332 
(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))