Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 7.1s

Alternatives: 3

Speedup: 1.2×

• 24.4% 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: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot z\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 20000000000:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* y z) z)))
   (if (<= t_0 -1e-24) t_0 (if (<= t_0 20000000000.0) (* 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 <= -1e-24) {
		tmp = t_0;
	} else if (t_0 <= 20000000000.0) {
		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 <= (-1d-24)) then
        tmp = t_0
    else if (t_0 <= 20000000000.0d0) 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 <= -1e-24) {
		tmp = t_0;
	} else if (t_0 <= 20000000000.0) {
		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 <= -1e-24:
		tmp = t_0
	elif t_0 <= 20000000000.0:
		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 <= -1e-24)
		tmp = t_0;
	elseif (t_0 <= 20000000000.0)
		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 <= -1e-24)
		tmp = t_0;
	elseif (t_0 <= 20000000000.0)
		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, -1e-24], t$95$0, If[LessEqual[t$95$0, 20000000000.0], N[(1.0 * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -9.99999999999999924e-25 or 2e10 < (*.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 84.2

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

   5. Applied rewrites 84.2%

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

   7. Applied rewrites 92.6%

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

   if -9.99999999999999924e-25 < (*.f64 (*.f64 y z) z) < 2e10

   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. flip-+ N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   4. Applied rewrites 55.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. mul-1-neg N/A

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

      22. lower-neg.f64 55.7

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

   6. Applied rewrites 55.7%

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 91.0%

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

Alternative 3: 51.4% 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. flip-+ N/A

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

   3. frac-2neg N/A

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

   4. div-inv N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. distribute-neg-in N/A

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

   8. neg-mul-1 N/A

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

   9. remove-double-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

4. Applied rewrites 44.3%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r* N/A

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

   20. lower-*.f64 N/A

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

   21. mul-1-neg N/A

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

   22. lower-neg.f64 42.5

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

6. Applied rewrites 42.5%

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

8. Applied rewrites 78.4%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 52.8%

    \[\leadsto \color{blue}{1} \cdot x \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))