Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 5.8s

Alternatives: 3

Speedup: 1.2×

• 24.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Alternative 2: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot z\right) \cdot z\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+93} \lor \neg \left(t\_0 \leq 5 \cdot 10^{-94}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (* y z) z)))
   (if (or (<= t_0 -2e+93) (not (<= t_0 5e-94))) t_0 (* -1.0 (- x)))))

C

double code(double x, double y, double z) {
	double t_0 = (y * z) * z;
	double tmp;
	if ((t_0 <= -2e+93) || !(t_0 <= 5e-94)) {
		tmp = t_0;
	} else {
		tmp = -1.0 * -x;
	}
	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+93)) .or. (.not. (t_0 <= 5d-94))) then
        tmp = t_0
    else
        tmp = (-1.0d0) * -x
    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+93) || !(t_0 <= 5e-94)) {
		tmp = t_0;
	} else {
		tmp = -1.0 * -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * z) * z
	tmp = 0
	if (t_0 <= -2e+93) or not (t_0 <= 5e-94):
		tmp = t_0
	else:
		tmp = -1.0 * -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * z) * z)
	tmp = 0.0
	if ((t_0 <= -2e+93) || !(t_0 <= 5e-94))
		tmp = t_0;
	else
		tmp = Float64(-1.0 * Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * z) * z;
	tmp = 0.0;
	if ((t_0 <= -2e+93) || ~((t_0 <= 5e-94)))
		tmp = t_0;
	else
		tmp = -1.0 * -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+93], N[Not[LessEqual[t$95$0, 5e-94]], $MachinePrecision]], t$95$0, N[(-1.0 * (-x)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -2.00000000000000009e93 or 4.9999999999999995e-94 < (*.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 x around 0

      \[\leadsto \color{blue}{y \cdot {z}^{2}} \]
   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 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 87.2%

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

   if -2.00000000000000009e93 < (*.f64 (*.f64 y z) z) < 4.9999999999999995e-94

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot {z}^{2}} \]
   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 11.9

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

   5. Applied rewrites 11.9%

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

   7. Applied rewrites 14.7%

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

   8. Taylor expanded in x around -inf

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

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 94.3

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

   10. Applied rewrites 94.3%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 88.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot z\right) \cdot z \leq -2 \cdot 10^{+93} \lor \neg \left(\left(y \cdot z\right) \cdot z \leq 5 \cdot 10^{-94}\right):\\ \;\;\;\;\left(y \cdot z\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.6% accurate, 1.8× speedup

Math

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

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 * Float64(-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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{y \cdot {z}^{2}} \]
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 46.5

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

5. Applied rewrites 46.5%

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

7. Applied rewrites 54.1%

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

8. Taylor expanded in x around -inf

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

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 84.1

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

10. Applied rewrites 84.1%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 47.5%

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

Reproduce

herbie shell --seed 2024317 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))