Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 5.9s

Alternatives: 3

Speedup: 1.0×

• 25.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -4.1 \cdot 10^{+17} \lor \neg \left(t\_0 \leq 8.5 \cdot 10^{+38}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))))
   (if (or (<= t_0 -4.1e+17) (not (<= t_0 8.5e+38))) t_0 x)))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if ((t_0 <= -4.1e+17) || !(t_0 <= 8.5e+38)) {
		tmp = t_0;
	} else {
		tmp = 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 = z * (y * z)
    if ((t_0 <= (-4.1d+17)) .or. (.not. (t_0 <= 8.5d+38))) then
        tmp = t_0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if ((t_0 <= -4.1e+17) || !(t_0 <= 8.5e+38)) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * z)
	tmp = 0
	if (t_0 <= -4.1e+17) or not (t_0 <= 8.5e+38):
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -4.1e+17) || !(t_0 <= 8.5e+38))
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y * z);
	tmp = 0.0;
	if ((t_0 <= -4.1e+17) || ~((t_0 <= 8.5e+38)))
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4.1e+17], N[Not[LessEqual[t$95$0, 8.5e+38]], $MachinePrecision]], t$95$0, x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -4.1e17 or 8.4999999999999997e38 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-*l* 82.8%

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

      3. fma-define 82.8%

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

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, z \cdot z, x\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 44.9%

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

      2. pow2 44.9%

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

      3. pow2 44.9%

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

   6. Applied egg-rr 44.9%

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

   7. Taylor expanded in z around -inf 55.5%

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

      1. mul-1-neg 55.5%

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

      2. *-commutative 55.5%

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

      3. distribute-rgt-neg-in 55.5%

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

   9. Simplified 55.5%

      \[\leadsto {\color{blue}{\left(z \cdot \left(-\sqrt{y}\right)\right)}}^{2} \]
   10. Step-by-step derivation

       1. *-commutative 55.5%

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

       2. unpow-prod-down 44.9%

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

       3. pow2 44.9%

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

       4. sqr-neg 44.9%

          \[\leadsto \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \cdot {z}^{2} \]

       5. add-sqr-sqrt 78.6%

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

       6. unpow2 78.6%

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

       7. associate-*r* 94.3%

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

   11. Applied egg-rr 94.3%

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

   if -4.1e17 < (*.f64 (*.f64 y z) z) < 8.4999999999999997e38

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*l* 96.9%

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

      3. fma-define 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in y around 0 90.9%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -4.1 \cdot 10^{+17} \lor \neg \left(z \cdot \left(y \cdot z\right) \leq 8.5 \cdot 10^{+38}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-*l* 89.7%

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

   3. fma-define 89.7%

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

3. Simplified 89.7%

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

5. Taylor expanded in y around 0 48.3%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 48.3%

   \[\leadsto x \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))