Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 4.5s

Alternatives: 5

Speedup: 1.0×

• 24.9% 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: 95.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{+152}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.05e+152) (+ x (* y (* z z))) (* z (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.05e+152) {
		tmp = x + (y * (z * z));
	} else {
		tmp = z * (y * 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 <= 1.05d+152) then
        tmp = x + (y * (z * z))
    else
        tmp = z * (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.05e+152) {
		tmp = x + (y * (z * z));
	} else {
		tmp = z * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.05e+152:
		tmp = x + (y * (z * z))
	else:
		tmp = z * (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.05e+152)
		tmp = Float64(x + Float64(y * Float64(z * z)));
	else
		tmp = Float64(z * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.05e+152)
		tmp = x + (y * (z * z));
	else
		tmp = z * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.05e+152], N[(x + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.0500000000000001e152

   1. Initial program 99.9%

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

      1. associate-*l* 95.1%

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

   3. Simplified 95.1%

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

   if 1.0500000000000001e152 < z

   1. Initial program 99.9%

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

      1. associate-*l* 84.5%

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

   3. Simplified 84.5%

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

      1. add-cbrt-cube 84.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(x + y \cdot \left(z \cdot z\right)\right) \cdot \left(x + y \cdot \left(z \cdot z\right)\right)\right) \cdot \left(x + y \cdot \left(z \cdot z\right)\right)}} \]

      2. pow3 84.5%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(x + y \cdot \left(z \cdot z\right)\right)}^{3}}} \]

      3. +-commutative 84.5%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(y \cdot \left(z \cdot z\right) + x\right)}}^{3}} \]

      4. associate-*r* 84.5%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(y \cdot z\right) \cdot z} + x\right)}^{3}} \]

      5. *-commutative 84.5%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{z \cdot \left(y \cdot z\right)} + x\right)}^{3}} \]

      6. fma-def 84.5%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(z, y \cdot z, x\right)\right)}}^{3}} \]

   5. Applied egg-rr 84.5%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(z, y \cdot z, x\right)\right)}^{3}}} \]

   6. Taylor expanded in z around inf 84.5%

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

      1. unpow2 84.5%

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

   8. Simplified 84.5%

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

      1. add-cube-cbrt 84.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot \left(z \cdot z\right)} \cdot \sqrt[3]{y \cdot \left(z \cdot z\right)}\right) \cdot \sqrt[3]{y \cdot \left(z \cdot z\right)}} \]

      2. pow3 84.5%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(z \cdot z\right)}\right)}^{3}} \]

      3. associate-*r* 99.8%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(y \cdot z\right) \cdot z}}\right)}^{3} \]

      4. *-commutative 99.8%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{z \cdot \left(y \cdot z\right)}}\right)}^{3} \]

   10. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{z \cdot \left(y \cdot z\right)}\right)}^{3}} \]
   11. Step-by-step derivation

       1. rem-cube-cbrt 99.9%

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

       2. associate-*r* 99.9%

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

   12. Applied egg-rr 99.9%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{+152}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 3: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 1.65e-25) x (* y (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.65e-25) {
		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 <= 1.65d-25) 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 <= 1.65e-25) {
		tmp = x;
	} else {
		tmp = y * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.65e-25:
		tmp = x
	else:
		tmp = y * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.65e-25)
		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 <= 1.65e-25)
		tmp = x;
	else
		tmp = y * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.65e-25], x, N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6499999999999999e-25

   1. Initial program 99.9%

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

      1. associate-*l* 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in x around inf 65.5%

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

   if 1.6499999999999999e-25 < z

   1. Initial program 99.9%

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

      1. associate-*l* 92.4%

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

   3. Simplified 92.4%

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

      1. add-cbrt-cube 64.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(x + y \cdot \left(z \cdot z\right)\right) \cdot \left(x + y \cdot \left(z \cdot z\right)\right)\right) \cdot \left(x + y \cdot \left(z \cdot z\right)\right)}} \]

      2. pow3 64.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(x + y \cdot \left(z \cdot z\right)\right)}^{3}}} \]

      3. +-commutative 64.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(y \cdot \left(z \cdot z\right) + x\right)}}^{3}} \]

      4. associate-*r* 64.3%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(y \cdot z\right) \cdot z} + x\right)}^{3}} \]

      5. *-commutative 64.3%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{z \cdot \left(y \cdot z\right)} + x\right)}^{3}} \]

      6. fma-def 64.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(z, y \cdot z, x\right)\right)}}^{3}} \]

   5. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(z, y \cdot z, x\right)\right)}^{3}}} \]

   6. Taylor expanded in z around inf 78.5%

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

      1. unpow2 78.5%

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

   8. Simplified 78.5%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 4: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.4e-25) x (* z (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e-25) {
		tmp = x;
	} else {
		tmp = z * (y * 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 <= 2.4d-25) then
        tmp = x
    else
        tmp = z * (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e-25) {
		tmp = x;
	} else {
		tmp = z * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.4e-25:
		tmp = x
	else:
		tmp = z * (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.4e-25)
		tmp = x;
	else
		tmp = Float64(z * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.4e-25)
		tmp = x;
	else
		tmp = z * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.4e-25], x, N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.40000000000000009e-25

   1. Initial program 99.9%

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

      1. associate-*l* 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in x around inf 65.5%

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

   if 2.40000000000000009e-25 < z

   1. Initial program 99.9%

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

      1. associate-*l* 92.4%

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

   3. Simplified 92.4%

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

      1. add-cbrt-cube 64.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(x + y \cdot \left(z \cdot z\right)\right) \cdot \left(x + y \cdot \left(z \cdot z\right)\right)\right) \cdot \left(x + y \cdot \left(z \cdot z\right)\right)}} \]

      2. pow3 64.3%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(x + y \cdot \left(z \cdot z\right)\right)}^{3}}} \]

      3. +-commutative 64.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(y \cdot \left(z \cdot z\right) + x\right)}}^{3}} \]

      4. associate-*r* 64.3%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(y \cdot z\right) \cdot z} + x\right)}^{3}} \]

      5. *-commutative 64.3%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{z \cdot \left(y \cdot z\right)} + x\right)}^{3}} \]

      6. fma-def 64.3%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(z, y \cdot z, x\right)\right)}}^{3}} \]

   5. Applied egg-rr 64.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(z, y \cdot z, x\right)\right)}^{3}}} \]

   6. Taylor expanded in z around inf 78.5%

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

      1. unpow2 78.5%

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

   8. Simplified 78.5%

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

      1. add-cube-cbrt 78.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{y \cdot \left(z \cdot z\right)} \cdot \sqrt[3]{y \cdot \left(z \cdot z\right)}\right) \cdot \sqrt[3]{y \cdot \left(z \cdot z\right)}} \]

      2. pow3 78.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{y \cdot \left(z \cdot z\right)}\right)}^{3}} \]

      3. associate-*r* 85.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(y \cdot z\right) \cdot z}}\right)}^{3} \]

      4. *-commutative 85.4%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{z \cdot \left(y \cdot z\right)}}\right)}^{3} \]

   10. Applied egg-rr 85.4%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{z \cdot \left(y \cdot z\right)}\right)}^{3}} \]
   11. Step-by-step derivation

       1. rem-cube-cbrt 86.1%

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

       2. associate-*r* 86.1%

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

   12. Applied egg-rr 86.1%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 5: 51.4% 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* 93.9%

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

3. Simplified 93.9%

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

4. Taylor expanded in x around inf 53.5%

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

5. Final simplification 53.5%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))