Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 7.6s

Alternatives: 3

Speedup: 1.2×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 86.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot z\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* y z) z)))
   (if (<= t_0 -5e-46) t_0 (if (<= t_0 5e-88) (* 1.0 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y * z) * z;
	double tmp;
	if (t_0 <= -5e-46) {
		tmp = t_0;
	} else if (t_0 <= 5e-88) {
		tmp = 1.0 * x;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (y * z) * z
    if (t_0 <= (-5d-46)) then
        tmp = t_0
    else if (t_0 <= 5d-88) then
        tmp = 1.0d0 * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * z) * z;
	double tmp;
	if (t_0 <= -5e-46) {
		tmp = t_0;
	} else if (t_0 <= 5e-88) {
		tmp = 1.0 * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * z) * z
	tmp = 0
	if t_0 <= -5e-46:
		tmp = t_0
	elif t_0 <= 5e-88:
		tmp = 1.0 * x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * z) * z)
	tmp = 0.0
	if (t_0 <= -5e-46)
		tmp = t_0;
	elseif (t_0 <= 5e-88)
		tmp = Float64(1.0 * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * z) * z;
	tmp = 0.0;
	if (t_0 <= -5e-46)
		tmp = t_0;
	elseif (t_0 <= 5e-88)
		tmp = 1.0 * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-46], t$95$0, If[LessEqual[t$95$0, 5e-88], N[(1.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -4.99999999999999992e-46 or 5.00000000000000009e-88 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{y \cdot {z}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot y} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{z}^{2} \cdot y} \]

      3. unpow2 N/A

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

      4. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 86.5%

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

   if -4.99999999999999992e-46 < (*.f64 (*.f64 y z) z) < 5.00000000000000009e-88

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. difference-of-squares N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower--.f64 54.8

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 54.8

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

   4. Applied rewrites 54.8%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-/l* N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{y \cdot {z}^{2}}{x}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \frac{y \cdot {z}^{2}}{x}\right) \cdot x} \]

      2. rgt-mult-inverse N/A

         \[\leadsto \left(\color{blue}{y \cdot \frac{1}{y}} + \frac{y \cdot {z}^{2}}{x}\right) \cdot x \]

      3. associate-/l* N/A

         \[\leadsto \left(y \cdot \frac{1}{y} + \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \cdot x \]

      4. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(\frac{1}{y} + \frac{{z}^{2}}{x}\right)\right)} \cdot x \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot \left(\frac{1}{y} + \frac{{z}^{2}}{x}\right)\right) \cdot x} \]

      6. +-commutative N/A

         \[\leadsto \left(y \cdot \color{blue}{\left(\frac{{z}^{2}}{x} + \frac{1}{y}\right)}\right) \cdot x \]

      7. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\frac{{z}^{2}}{x} \cdot y + \frac{1}{y} \cdot y\right)} \cdot x \]

      8. lft-mult-inverse N/A

         \[\leadsto \left(\frac{{z}^{2}}{x} \cdot y + \color{blue}{1}\right) \cdot x \]

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 95.3

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

   10. Applied rewrites 95.3%

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

   11. Taylor expanded in z around 0

       \[\leadsto 1 \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 92.9%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 50.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z) {
	return 1.0 * x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 * x
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 * x;
}

Python

def code(x, y, z):
	return 1.0 * x

Julia

function code(x, y, z)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lower-/.f64 N/A

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

   5. difference-of-squares N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower--.f64 39.1

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

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 39.1

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

4. Applied rewrites 39.1%

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

5. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. unpow2 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f64 88.4

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

7. Applied rewrites 88.4%

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

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{y \cdot {z}^{2}}{x}\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(1 + \frac{y \cdot {z}^{2}}{x}\right) \cdot x} \]

   2. rgt-mult-inverse N/A

      \[\leadsto \left(\color{blue}{y \cdot \frac{1}{y}} + \frac{y \cdot {z}^{2}}{x}\right) \cdot x \]

   3. associate-/l* N/A

      \[\leadsto \left(y \cdot \frac{1}{y} + \color{blue}{y \cdot \frac{{z}^{2}}{x}}\right) \cdot x \]

   4. distribute-lft-in N/A

      \[\leadsto \color{blue}{\left(y \cdot \left(\frac{1}{y} + \frac{{z}^{2}}{x}\right)\right)} \cdot x \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot \left(\frac{1}{y} + \frac{{z}^{2}}{x}\right)\right) \cdot x} \]

   6. +-commutative N/A

      \[\leadsto \left(y \cdot \color{blue}{\left(\frac{{z}^{2}}{x} + \frac{1}{y}\right)}\right) \cdot x \]

   7. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\frac{{z}^{2}}{x} \cdot y + \frac{1}{y} \cdot y\right)} \cdot x \]

   8. lft-mult-inverse N/A

      \[\leadsto \left(\frac{{z}^{2}}{x} \cdot y + \color{blue}{1}\right) \cdot x \]

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 85.2

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

10. Applied rewrites 85.2%

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

11. Taylor expanded in z around 0

    \[\leadsto 1 \cdot x \]
12. Step-by-step derivation

13. Applied rewrites 49.4%

    \[\leadsto 1 \cdot x \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))