Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 4.9s

Alternatives: 6

Speedup: 1.0×

• 26.0% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l* 92.7%

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

3. Simplified 92.7%

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

   1. +-commutative 92.7%

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

   2. associate-*r* 99.9%

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

   3. fma-def 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+165} \lor \neg \left(z \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.5e+165) (not (<= z 1.32e+154)))
   (* z (* y z))
   (+ x (* y (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e+165) || !(z <= 1.32e+154)) {
		tmp = z * (y * z);
	} else {
		tmp = x + (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 <= (-6.5d+165)) .or. (.not. (z <= 1.32d+154))) then
        tmp = z * (y * z)
    else
        tmp = x + (y * (z * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5e+165) || !(z <= 1.32e+154)) {
		tmp = z * (y * z);
	} else {
		tmp = x + (y * (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -6.5e+165) or not (z <= 1.32e+154):
		tmp = z * (y * z)
	else:
		tmp = x + (y * (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.5e+165) || !(z <= 1.32e+154))
		tmp = Float64(z * Float64(y * z));
	else
		tmp = Float64(x + Float64(y * Float64(z * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -6.5e+165) || ~((z <= 1.32e+154)))
		tmp = z * (y * z);
	else
		tmp = x + (y * (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.5e+165], N[Not[LessEqual[z, 1.32e+154]], $MachinePrecision]], N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.4999999999999999e165 or 1.31999999999999998e154 < z

   1. Initial program 99.9%

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

      1. associate-*l* 76.5%

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

   3. Simplified 76.5%

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

      1. +-commutative 76.5%

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

      2. associate-*r* 99.9%

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

      3. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around inf 76.5%

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

      1. unpow2 76.5%

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

      2. associate-*r* 99.7%

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

      3. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   if -6.4999999999999999e165 < z < 1.31999999999999998e154

   1. Initial program 99.9%

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

      1. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+165} \lor \neg \left(z \leq 1.32 \cdot 10^{+154}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 3: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -720 \lor \neg \left(z \leq 4 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -720.0) (not (<= z 4e-7))) (* y (* z z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -720.0) || !(z <= 4e-7)) {
		tmp = y * (z * z);
	} 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) :: tmp
    if ((z <= (-720.0d0)) .or. (.not. (z <= 4d-7))) then
        tmp = y * (z * z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -720.0) || !(z <= 4e-7)) {
		tmp = y * (z * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -720.0) or not (z <= 4e-7):
		tmp = y * (z * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -720.0) || !(z <= 4e-7))
		tmp = Float64(y * Float64(z * z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -720.0) || ~((z <= 4e-7)))
		tmp = y * (z * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -720.0], N[Not[LessEqual[z, 4e-7]], $MachinePrecision]], N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -720 or 3.9999999999999998e-7 < z

   1. Initial program 99.9%

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

      1. associate-*l* 86.3%

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

   3. Simplified 86.3%

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

      1. +-commutative 86.3%

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

      2. associate-*r* 99.9%

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

      3. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around inf 75.0%

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

      1. unpow2 75.0%

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

   8. Simplified 75.0%

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

   if -720 < z < 3.9999999999999998e-7

   1. Initial program 100.0%

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

      1. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around inf 87.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -720 \lor \neg \left(z \leq 4 \cdot 10^{-7}\right):\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -34 \lor \neg \left(z \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -34.0) (not (<= z 4.8e-6))) (* z (* y z)) x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -34.0) || !(z <= 4.8e-6)) {
		tmp = z * (y * z);
	} 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) :: tmp
    if ((z <= (-34.0d0)) .or. (.not. (z <= 4.8d-6))) then
        tmp = z * (y * z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -34.0) || !(z <= 4.8e-6)) {
		tmp = z * (y * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -34.0) or not (z <= 4.8e-6):
		tmp = z * (y * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -34.0) || !(z <= 4.8e-6))
		tmp = Float64(z * Float64(y * z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -34.0) || ~((z <= 4.8e-6)))
		tmp = z * (y * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -34.0], N[Not[LessEqual[z, 4.8e-6]], $MachinePrecision]], N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -34 or 4.7999999999999998e-6 < z

   1. Initial program 99.9%

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

      1. associate-*l* 86.3%

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

   3. Simplified 86.3%

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

      1. +-commutative 86.3%

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

      2. associate-*r* 99.9%

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

      3. fma-def 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in y around inf 75.0%

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

      1. unpow2 75.0%

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

      2. associate-*r* 88.6%

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

      3. *-commutative 88.6%

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

   8. Simplified 88.6%

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

   if -34 < z < 4.7999999999999998e-6

   1. Initial program 100.0%

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

      1. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around inf 87.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -34 \lor \neg \left(z \leq 4.8 \cdot 10^{-6}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 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. Final simplification 99.9%

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

Alternative 6: 49.9% 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* 92.7%

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

3. Simplified 92.7%

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

4. Taylor expanded in x around inf 49.8%

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

5. Final simplification 49.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023195 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))