Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 3

Speedup: 1.2×

• 24.3% 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: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot z\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* y z) z)))
   (if (<= t_0 -2e+23) t_0 (if (<= t_0 5.0) (* (- x) -1.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y * z) * z;
	double tmp;
	if (t_0 <= -2e+23) {
		tmp = t_0;
	} else if (t_0 <= 5.0) {
		tmp = -x * -1.0;
	} 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 <= (-2d+23)) then
        tmp = t_0
    else if (t_0 <= 5.0d0) then
        tmp = -x * (-1.0d0)
    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 <= -2e+23) {
		tmp = t_0;
	} else if (t_0 <= 5.0) {
		tmp = -x * -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * z) * z
	tmp = 0
	if t_0 <= -2e+23:
		tmp = t_0
	elif t_0 <= 5.0:
		tmp = -x * -1.0
	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 <= -2e+23)
		tmp = t_0;
	elseif (t_0 <= 5.0)
		tmp = Float64(Float64(-x) * -1.0);
	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 <= -2e+23)
		tmp = t_0;
	elseif (t_0 <= 5.0)
		tmp = -x * -1.0;
	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, -2e+23], t$95$0, If[LessEqual[t$95$0, 5.0], N[((-x) * -1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -1.9999999999999998e23 or 5 < (*.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 77.4

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

   5. Applied rewrites 77.4%

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

   7. Applied rewrites 90.8%

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

   if -1.9999999999999998e23 < (*.f64 (*.f64 y z) z) < 5

   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. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

      7. lift-+.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.7

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   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. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. lower-neg.f64 N/A

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

      9. +-commutative N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 96.5

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

   7. Applied rewrites 96.5%

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

   8. Taylor expanded in z around 0

      \[\leadsto -1 \cdot \left(-\color{blue}{x}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.1%

       \[\leadsto -1 \cdot \left(-\color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot z\right) \cdot z \leq -2 \cdot 10^{+23}:\\ \;\;\;\;\left(y \cdot z\right) \cdot z\\ \mathbf{elif}\;\left(y \cdot z\right) \cdot z \leq 5:\\ \;\;\;\;\left(-x\right) \cdot -1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) \cdot -1 \end{array} \]

FPCore

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

C

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

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 * (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[((-x) * -1.0), $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. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

   7. lift-+.f64 N/A

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

   8. inv-pow N/A

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

   9. lower-pow.f64 99.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.8

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

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. mul-1-neg N/A

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

   6. distribute-neg-in N/A

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

   7. +-commutative N/A

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

   8. lower-neg.f64 N/A

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

   9. +-commutative N/A

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

   10. associate-/l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f64 87.4

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

7. Applied rewrites 87.4%

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

8. Taylor expanded in z around 0

   \[\leadsto -1 \cdot \left(-\color{blue}{x}\right) \]
9. Step-by-step derivation

10. Applied rewrites 54.3%

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

11. Final simplification 54.3%

    \[\leadsto \left(-x\right) \cdot -1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024259 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))