Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 3

Speedup: N/A×

• 24.8% 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: 50.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(y * z), $MachinePrecision] * z), $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 99.7

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

   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 z around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 47.9

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

7. Applied rewrites 47.9%

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

8. Final simplification 47.9%

   \[\leadsto \left(y \cdot z\right) \cdot z \]
9. Add Preprocessing

Alternative 3: 45.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[x + \left(y \cdot z\right) \cdot z \]
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 43.4

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

5. Applied rewrites 43.4%

   \[\leadsto \color{blue}{\left(z \cdot z\right) \cdot y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024244 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))