Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 5

Speedup: 1.0×

• 25.2% 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. Final simplification 99.9%

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

Alternative 2: 76.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-14} \lor \neg \left(z \leq -1.32 \cdot 10^{-65} \lor \neg \left(z \leq -1.4 \cdot 10^{-116}\right) \land z \leq 2.2 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.3e-14)
         (not
          (or (<= z -1.32e-65) (and (not (<= z -1.4e-116)) (<= z 2.2e+15)))))
   (* y (* z z))
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.3e-14) || !((z <= -1.32e-65) || (!(z <= -1.4e-116) && (z <= 2.2e+15)))) {
		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 <= (-1.3d-14)) .or. (.not. (z <= (-1.32d-65)) .or. (.not. (z <= (-1.4d-116))) .and. (z <= 2.2d+15))) 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 <= -1.3e-14) || !((z <= -1.32e-65) || (!(z <= -1.4e-116) && (z <= 2.2e+15)))) {
		tmp = y * (z * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.3e-14) or not ((z <= -1.32e-65) or (not (z <= -1.4e-116) and (z <= 2.2e+15))):
		tmp = y * (z * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.3e-14) || !((z <= -1.32e-65) || (!(z <= -1.4e-116) && (z <= 2.2e+15))))
		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 <= -1.3e-14) || ~(((z <= -1.32e-65) || (~((z <= -1.4e-116)) && (z <= 2.2e+15)))))
		tmp = y * (z * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.3e-14], N[Not[Or[LessEqual[z, -1.32e-65], And[N[Not[LessEqual[z, -1.4e-116]], $MachinePrecision], LessEqual[z, 2.2e+15]]]], $MachinePrecision]], N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999998e-14 or -1.32e-65 < z < -1.3999999999999999e-116 or 2.2e15 < z

   1. Initial program 99.9%

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

      1. associate-*l* 90.6%

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

   3. Simplified 90.6%

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

      1. +-commutative 90.6%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. add-sqr-sqrt 51.4%

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

      5. associate-*r* 51.4%

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

      6. fma-def 51.4%

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

   5. Applied egg-rr 51.4%

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

   6. Taylor expanded in x around 0 77.3%

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

      1. unpow2 77.3%

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

      2. associate-*l* 83.9%

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

      3. *-commutative 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in z around 0 77.3%

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

       1. unpow2 77.3%

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

   11. Simplified 77.3%

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

   if -1.29999999999999998e-14 < z < -1.32e-65 or -1.3999999999999999e-116 < z < 2.2e15

   1. Initial program 100.0%

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

      1. associate-*l* 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 87.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-14} \lor \neg \left(z \leq -1.32 \cdot 10^{-65} \lor \neg \left(z \leq -1.4 \cdot 10^{-116}\right) \land z \leq 2.2 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-12} \lor \neg \left(z \leq -1.05 \cdot 10^{-65} \lor \neg \left(z \leq -1.4 \cdot 10^{-116}\right) \land z \leq 2.2 \cdot 10^{+15}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.1e-12)
         (not
          (or (<= z -1.05e-65) (and (not (<= z -1.4e-116)) (<= z 2.2e+15)))))
   (* z (* y z))
   x))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.1e-12) || !((z <= -1.05e-65) || (!(z <= -1.4e-116) && (z <= 2.2e+15)))) {
		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 <= (-1.1d-12)) .or. (.not. (z <= (-1.05d-65)) .or. (.not. (z <= (-1.4d-116))) .and. (z <= 2.2d+15))) 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 <= -1.1e-12) || !((z <= -1.05e-65) || (!(z <= -1.4e-116) && (z <= 2.2e+15)))) {
		tmp = z * (y * z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.1e-12) or not ((z <= -1.05e-65) or (not (z <= -1.4e-116) and (z <= 2.2e+15))):
		tmp = z * (y * z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.1e-12) || !((z <= -1.05e-65) || (!(z <= -1.4e-116) && (z <= 2.2e+15))))
		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 <= -1.1e-12) || ~(((z <= -1.05e-65) || (~((z <= -1.4e-116)) && (z <= 2.2e+15)))))
		tmp = z * (y * z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.1e-12], N[Not[Or[LessEqual[z, -1.05e-65], And[N[Not[LessEqual[z, -1.4e-116]], $MachinePrecision], LessEqual[z, 2.2e+15]]]], $MachinePrecision]], N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.09999999999999996e-12 or -1.05000000000000001e-65 < z < -1.3999999999999999e-116 or 2.2e15 < z

   1. Initial program 99.9%

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

      1. associate-*l* 90.6%

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

   3. Simplified 90.6%

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

      1. +-commutative 90.6%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. add-sqr-sqrt 51.4%

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

      5. associate-*r* 51.4%

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

      6. fma-def 51.4%

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

   5. Applied egg-rr 51.4%

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

   6. Taylor expanded in x around 0 77.3%

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

      1. unpow2 77.3%

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

      2. associate-*l* 83.9%

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

      3. *-commutative 83.9%

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

   8. Simplified 83.9%

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

   if -1.09999999999999996e-12 < z < -1.05000000000000001e-65 or -1.3999999999999999e-116 < z < 2.2e15

   1. Initial program 100.0%

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

      1. associate-*l* 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 87.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-12} \lor \neg \left(z \leq -1.05 \cdot 10^{-65} \lor \neg \left(z \leq -1.4 \cdot 10^{-116}\right) \land z \leq 2.2 \cdot 10^{+15}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+126}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.6e+126) (* z (* y z)) (+ x (* y (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.6e+126) {
		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 <= (-7.6d+126)) 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 <= -7.6e+126) {
		tmp = z * (y * z);
	} else {
		tmp = x + (y * (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.6e+126:
		tmp = z * (y * z)
	else:
		tmp = x + (y * (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.6e+126)
		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 <= -7.6e+126)
		tmp = z * (y * z);
	else
		tmp = x + (y * (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.6e+126], 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 < -7.60000000000000033e126

   1. Initial program 100.0%

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

      1. associate-*l* 82.5%

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

   3. Simplified 82.5%

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

      1. +-commutative 82.5%

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

      2. associate-*r* 100.0%

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

      3. *-commutative 100.0%

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

      4. add-sqr-sqrt 65.5%

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

      5. associate-*r* 65.6%

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

      6. fma-def 65.5%

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

   5. Applied egg-rr 65.5%

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

   6. Taylor expanded in x around 0 82.5%

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

      1. unpow2 82.5%

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

      2. associate-*l* 98.3%

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

      3. *-commutative 98.3%

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

   8. Simplified 98.3%

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

   if -7.60000000000000033e126 < z

   1. Initial program 99.9%

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

      1. associate-*l* 96.2%

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

   3. Simplified 96.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+126}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 5: 50.3% 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* 94.5%

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

3. Simplified 94.5%

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

4. Taylor expanded in x around inf 51.7%

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

5. Final simplification 51.7%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023185 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))