Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 7.9s

Alternatives: 5

Speedup: 1.0×

• 25.4% 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%

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

   3. *-commutative 99.9%

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

   4. fma-def 99.9%

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

   5. *-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 87.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot y\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+31} \lor \neg \left(t_0 \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* z y))))
   (if (or (<= t_0 -5e+31) (not (<= t_0 5e-9))) t_0 x)))

C

double code(double x, double y, double z) {
	double t_0 = z * (z * y);
	double tmp;
	if ((t_0 <= -5e+31) || !(t_0 <= 5e-9)) {
		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 <= (-5d+31)) .or. (.not. (t_0 <= 5d-9))) 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 <= -5e+31) || !(t_0 <= 5e-9)) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (z * y)
	tmp = 0
	if (t_0 <= -5e+31) or not (t_0 <= 5e-9):
		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 <= -5e+31) || !(t_0 <= 5e-9))
		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 <= -5e+31) || ~((t_0 <= 5e-9)))
		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, -5e+31], N[Not[LessEqual[t$95$0, 5e-9]], $MachinePrecision]], t$95$0, x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -5.00000000000000027e31 or 5.0000000000000001e-9 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.8%

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

      1. associate-*l* 84.7%

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

   3. Simplified 84.7%

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

      1. flip-+ 15.7%

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

      2. clear-num 15.7%

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

      3. clear-num 15.7%

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

      4. flip-+ 84.7%

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

      5. +-commutative 84.7%

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

      6. fma-udef 84.7%

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

      7. pow2 84.7%

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

   5. Applied egg-rr 84.7%

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

   6. Taylor expanded in y around inf 76.4%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{y \cdot {z}^{2}}}} \]
   7. Step-by-step derivation

      1. remove-double-div 76.5%

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

      2. unpow2 76.5%

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

      3. associate-*r* 90.7%

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

   8. Applied egg-rr 90.7%

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

   if -5.00000000000000027e31 < (*.f64 (*.f64 y z) z) < 5.0000000000000001e-9

   1. Initial program 99.9%

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

      1. associate-*l* 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in x around inf 87.1%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(z \cdot y\right) \leq -5 \cdot 10^{+31} \lor \neg \left(z \cdot \left(z \cdot y\right) \leq 5 \cdot 10^{-9}\right):\\ \;\;\;\;z \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.35e+154) (+ x (* y (* z z))) (* z (* z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.35e+154) {
		tmp = x + (y * (z * z));
	} else {
		tmp = z * (z * y);
	}
	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 <= 1.35d+154) then
        tmp = x + (y * (z * z))
    else
        tmp = z * (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.35e+154) {
		tmp = x + (y * (z * z));
	} else {
		tmp = z * (z * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.35e+154:
		tmp = x + (y * (z * z))
	else:
		tmp = z * (z * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.35e+154)
		tmp = Float64(x + Float64(y * Float64(z * z)));
	else
		tmp = Float64(z * Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.35e+154)
		tmp = x + (y * (z * z));
	else
		tmp = z * (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.35e+154], N[(x + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. associate-*l* 94.2%

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

   3. Simplified 94.2%

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

   if 1.35000000000000003e154 < z

   1. Initial program 99.8%

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

      1. associate-*l* 66.6%

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

   3. Simplified 66.6%

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

      1. flip-+ 0.0%

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

      2. clear-num 0.0%

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

      3. clear-num 0.0%

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

      4. flip-+ 66.6%

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

      5. +-commutative 66.6%

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

      6. fma-udef 66.6%

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

      7. pow2 66.6%

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

   5. Applied egg-rr 66.6%

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

   6. Taylor expanded in y around inf 66.6%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{y \cdot {z}^{2}}}} \]
   7. Step-by-step derivation

      1. remove-double-div 66.6%

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

      2. unpow2 66.6%

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

      3. associate-*r* 96.3%

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

   8. Applied egg-rr 96.3%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot y\right)\\ \end{array} \]

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. Final simplification 99.9%

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

Alternative 5: 49.6% 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. associate-*l* 91.2%

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

3. Simplified 91.2%

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

4. Taylor expanded in x around inf 51.9%

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

5. Final simplification 51.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))