Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

Alternatives: 5

Speedup: 1.0×

• 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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. *-commutative 99.9%

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

   3. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-17} \lor \neg \left(t\_0 \leq 4 \cdot 10^{-43}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z y))))
   (if (or (<= t_0 -1e-17) (not (<= t_0 4e-43))) t_0 x)))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * y);
	double tmp;
	if ((t_0 <= -1e-17) || !(t_0 <= 4e-43)) {
		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 * (z * y)
    if ((t_0 <= (-1d-17)) .or. (.not. (t_0 <= 4d-43))) 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 * (z * y);
	double tmp;
	if ((t_0 <= -1e-17) || !(t_0 <= 4e-43)) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (z * y)
	tmp = 0
	if (t_0 <= -1e-17) or not (t_0 <= 4e-43):
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(z * y))
	tmp = 0.0
	if ((t_0 <= -1e-17) || !(t_0 <= 4e-43))
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (z * y);
	tmp = 0.0;
	if ((t_0 <= -1e-17) || ~((t_0 <= 4e-43)))
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-17], N[Not[LessEqual[t$95$0, 4e-43]], $MachinePrecision]], t$95$0, x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -1.00000000000000007e-17 or 4.00000000000000031e-43 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. add-sqr-sqrt 43.4%

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

      3. associate-*r* 43.4%

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

      4. fma-define 43.4%

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

   4. Applied egg-rr 43.4%

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

   5. Taylor expanded in y around inf 80.9%

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

      1. add-sqr-sqrt 41.2%

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

      2. pow2 41.2%

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

      3. *-commutative 41.2%

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

      4. sqrt-prod 41.2%

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

      5. unpow2 41.2%

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

      6. sqrt-prod 19.3%

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

      7. add-sqr-sqrt 45.6%

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

   7. Applied egg-rr 45.6%

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

      1. unpow2 45.6%

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

      2. swap-sqr 41.1%

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

      3. add-sqr-sqrt 80.9%

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

      4. *-commutative 80.9%

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

      5. associate-*r* 91.3%

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

   9. Applied egg-rr 91.3%

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

   if -1.00000000000000007e-17 < (*.f64 (*.f64 y z) z) < 4.00000000000000031e-43

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 90.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot y\right) \leq -1 \cdot 10^{-17} \lor \neg \left(z \cdot \left(z \cdot y\right) \leq 4 \cdot 10^{-43}\right):\\ \;\;\;\;z \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 8e+31) x (* y (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 8e+31) {
		tmp = x;
	} else {
		tmp = y * (z * z);
	}
	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) :: tmp
    if (z <= 8d+31) then
        tmp = x
    else
        tmp = y * (z * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 8e+31) {
		tmp = x;
	} else {
		tmp = y * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 8e+31:
		tmp = x
	else:
		tmp = y * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 8e+31)
		tmp = x;
	else
		tmp = Float64(y * Float64(z * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 8e+31)
		tmp = x;
	else
		tmp = y * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 8e+31], x, N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.9999999999999997e31

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 57.8%

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

   if 7.9999999999999997e31 < z

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. add-sqr-sqrt 99.8%

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

      3. associate-*r* 99.8%

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

      4. fma-define 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf 85.6%

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

      1. unpow2 85.6%

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

   7. Applied egg-rr 85.6%

      \[\leadsto y \cdot \color{blue}{\left(z \cdot z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 5: 51.2% 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. Add Preprocessing

3. Taylor expanded in x around inf 46.9%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))