Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 6.3s

Alternatives: 4

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: 63.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{-85} \lor \neg \left(z \leq 9.2 \cdot 10^{-46}\right) \land z \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 1.3e-85) (and (not (<= z 9.2e-46)) (<= z 1.9e+36)))
   x
   (* y (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 1.3e-85) || (!(z <= 9.2e-46) && (z <= 1.9e+36))) {
		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 <= 1.3d-85) .or. (.not. (z <= 9.2d-46)) .and. (z <= 1.9d+36)) 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 <= 1.3e-85) || (!(z <= 9.2e-46) && (z <= 1.9e+36))) {
		tmp = x;
	} else {
		tmp = y * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 1.3e-85) or (not (z <= 9.2e-46) and (z <= 1.9e+36)):
		tmp = x
	else:
		tmp = y * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 1.3e-85) || (!(z <= 9.2e-46) && (z <= 1.9e+36)))
		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 <= 1.3e-85) || (~((z <= 9.2e-46)) && (z <= 1.9e+36)))
		tmp = x;
	else
		tmp = y * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 1.3e-85], And[N[Not[LessEqual[z, 9.2e-46]], $MachinePrecision], LessEqual[z, 1.9e+36]]], x, N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.30000000000000006e-85 or 9.1999999999999997e-46 < z < 1.90000000000000012e36

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 58.5%

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

   if 1.30000000000000006e-85 < z < 9.1999999999999997e-46 or 1.90000000000000012e36 < 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 99.7%

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

      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 80.5%

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

      1. unpow2 80.5%

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

   7. Applied egg-rr 80.5%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.3 \cdot 10^{-85} \lor \neg \left(z \leq 9.2 \cdot 10^{-46}\right) \land z \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 51.0% 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 47.7%

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

Reproduce

herbie shell --seed 2024146 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))