Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 5.7s

Alternatives: 5

Speedup: 1.0×

• 24.7% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{+153}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.7e+153) (+ x (* y (* z z))) (* z (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.7e+153) {
		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 <= 2.7d+153) 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 <= 2.7e+153) {
		tmp = x + (y * (z * z));
	} else {
		tmp = z * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.7e+153:
		tmp = x + (y * (z * z))
	else:
		tmp = z * (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.7e+153)
		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 <= 2.7e+153)
		tmp = x + (y * (z * z));
	else
		tmp = z * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.7e+153], 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 < 2.7000000000000001e153

   1. Initial program 99.9%

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

      1. associate-*l* 95.7%

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

   3. Simplified 95.7%

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

   if 2.7000000000000001e153 < z

   1. Initial program 99.9%

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

      1. associate-*l* 73.1%

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

   3. Simplified 73.1%

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

      1. add-cbrt-cube 73.1%

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

      2. pow3 73.1%

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

      3. add-sqr-sqrt 46.5%

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

      4. unpow-prod-down 46.5%

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

      5. pow-prod-up 46.5%

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

      6. *-commutative 46.5%

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

      7. sqrt-prod 46.5%

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

      8. sqrt-prod 46.5%

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

      9. add-sqr-sqrt 46.5%

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

      10. metadata-eval 46.5%

          \[\leadsto x + \sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{\color{blue}{6}}} \]

   5. Applied egg-rr 46.5%

      \[\leadsto x + \color{blue}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}}} \]
   6. Step-by-step derivation

      1. +-commutative 46.5%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} + x} \]

      2. flip-+ 0.0%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} \cdot \sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x}} \]

      3. pow2 0.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}}\right)}^{2}} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      4. pow1/3 0.0%

         \[\leadsto \frac{{\color{blue}{\left({\left({\left(z \cdot \sqrt{y}\right)}^{6}\right)}^{0.3333333333333333}\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      5. pow-pow 0.0%

         \[\leadsto \frac{{\color{blue}{\left({\left(z \cdot \sqrt{y}\right)}^{\left(6 \cdot 0.3333333333333333\right)}\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      6. metadata-eval 0.0%

         \[\leadsto \frac{{\left({\left(z \cdot \sqrt{y}\right)}^{\color{blue}{2}}\right)}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      7. pow2 0.0%

         \[\leadsto \frac{{\color{blue}{\left(\left(z \cdot \sqrt{y}\right) \cdot \left(z \cdot \sqrt{y}\right)\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      8. swap-sqr 0.0%

         \[\leadsto \frac{{\color{blue}{\left(\left(z \cdot z\right) \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      9. add-sqr-sqrt 0.0%

         \[\leadsto \frac{{\left(\left(z \cdot z\right) \cdot \color{blue}{y}\right)}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      10. associate-*r* 0.0%

          \[\leadsto \frac{{\color{blue}{\left(z \cdot \left(z \cdot y\right)\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      11. pow1/3 0.0%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{{\left({\left(z \cdot \sqrt{y}\right)}^{6}\right)}^{0.3333333333333333}} - x} \]

      12. pow-pow 0.6%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{{\left(z \cdot \sqrt{y}\right)}^{\left(6 \cdot 0.3333333333333333\right)}} - x} \]

      13. metadata-eval 0.6%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{{\left(z \cdot \sqrt{y}\right)}^{\color{blue}{2}} - x} \]

      14. pow2 0.6%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{\left(z \cdot \sqrt{y}\right) \cdot \left(z \cdot \sqrt{y}\right)} - x} \]

      15. swap-sqr 0.0%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{\left(z \cdot z\right) \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)} - x} \]

      16. add-sqr-sqrt 0.1%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\left(z \cdot z\right) \cdot \color{blue}{y} - x} \]

      17. associate-*r* 5.9%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{z \cdot \left(z \cdot y\right)} - x} \]

   7. Applied egg-rr 5.9%

      \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{z \cdot \left(z \cdot y\right) - x}} \]
   8. Step-by-step derivation

      1. unpow2 5.9%

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

      2. associate-*r* 1.7%

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

      3. associate-*r* 1.7%

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

   9. Applied egg-rr 1.7%

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

   10. Taylor expanded in z around inf 73.1%

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

       1. unpow2 73.1%

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

       2. associate-*r* 91.9%

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

       3. *-commutative 91.9%

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

       4. associate-*r* 91.9%

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

   12. Simplified 91.9%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{+153}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 3: 63.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 4.2e+76) x (* y (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 4.2e+76) {
		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 <= 4.2d+76) 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 <= 4.2e+76) {
		tmp = x;
	} else {
		tmp = y * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 4.2e+76:
		tmp = x
	else:
		tmp = y * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 4.2e+76)
		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 <= 4.2e+76)
		tmp = x;
	else
		tmp = y * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 4.2e+76], x, N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.20000000000000013e76

   1. Initial program 99.9%

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

      1. associate-*l* 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in x around inf 61.5%

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

   if 4.20000000000000013e76 < z

   1. Initial program 99.9%

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

      1. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

      1. add-cbrt-cube 69.2%

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

      2. pow3 69.2%

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

      3. add-sqr-sqrt 37.1%

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

      4. unpow-prod-down 37.1%

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

      5. pow-prod-up 37.1%

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

      6. *-commutative 37.1%

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

      7. sqrt-prod 37.1%

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

      8. sqrt-prod 37.1%

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

      9. add-sqr-sqrt 37.1%

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

      10. metadata-eval 37.1%

          \[\leadsto x + \sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{\color{blue}{6}}} \]

   5. Applied egg-rr 37.1%

      \[\leadsto x + \color{blue}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}}} \]
   6. Step-by-step derivation

      1. pow1/3 37.1%

         \[\leadsto x + \color{blue}{{\left({\left(z \cdot \sqrt{y}\right)}^{6}\right)}^{0.3333333333333333}} \]

      2. pow-pow 49.9%

         \[\leadsto x + \color{blue}{{\left(z \cdot \sqrt{y}\right)}^{\left(6 \cdot 0.3333333333333333\right)}} \]

      3. metadata-eval 49.9%

         \[\leadsto x + {\left(z \cdot \sqrt{y}\right)}^{\color{blue}{2}} \]

      4. pow2 49.9%

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

      5. swap-sqr 44.8%

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

      6. add-sqr-sqrt 82.0%

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

      7. associate-*r* 99.9%

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

      8. *-commutative 99.9%

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

      9. add-cube-cbrt 99.4%

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

      10. unpow3 99.3%

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

      11. flip-+ 15.5%

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

      12. frac-2neg 15.5%

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

   7. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{\frac{-\left(x \cdot x - {\left(z \cdot \left(z \cdot y\right)\right)}^{2}\right)}{-\left(x - z \cdot \left(z \cdot y\right)\right)}} \]
   8. Step-by-step derivation

      1. neg-sub0 15.8%

         \[\leadsto \frac{\color{blue}{0 - \left(x \cdot x - {\left(z \cdot \left(z \cdot y\right)\right)}^{2}\right)}}{-\left(x - z \cdot \left(z \cdot y\right)\right)} \]

      2. associate--r- 15.8%

         \[\leadsto \frac{\color{blue}{\left(0 - x \cdot x\right) + {\left(z \cdot \left(z \cdot y\right)\right)}^{2}}}{-\left(x - z \cdot \left(z \cdot y\right)\right)} \]

      3. neg-sub0 15.8%

         \[\leadsto \frac{\color{blue}{\left(-x \cdot x\right)} + {\left(z \cdot \left(z \cdot y\right)\right)}^{2}}{-\left(x - z \cdot \left(z \cdot y\right)\right)} \]

      4. +-commutative 15.8%

         \[\leadsto \frac{\color{blue}{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} + \left(-x \cdot x\right)}}{-\left(x - z \cdot \left(z \cdot y\right)\right)} \]

      5. sub-neg 15.8%

         \[\leadsto \frac{\color{blue}{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}}{-\left(x - z \cdot \left(z \cdot y\right)\right)} \]

      6. unpow2 15.8%

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

      7. associate-*r* 13.0%

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

      8. associate-*r* 13.0%

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

      9. swap-sqr 1.1%

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

      10. associate-*r* 1.1%

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

      11. unpow3 1.1%

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

      12. pow-plus 1.1%

          \[\leadsto \frac{\color{blue}{{z}^{\left(3 + 1\right)}} \cdot \left(y \cdot y\right) - x \cdot x}{-\left(x - z \cdot \left(z \cdot y\right)\right)} \]

      13. metadata-eval 1.1%

          \[\leadsto \frac{{z}^{\color{blue}{4}} \cdot \left(y \cdot y\right) - x \cdot x}{-\left(x - z \cdot \left(z \cdot y\right)\right)} \]

      14. neg-sub0 1.1%

          \[\leadsto \frac{{z}^{4} \cdot \left(y \cdot y\right) - x \cdot x}{\color{blue}{0 - \left(x - z \cdot \left(z \cdot y\right)\right)}} \]

      15. associate--r- 1.1%

          \[\leadsto \frac{{z}^{4} \cdot \left(y \cdot y\right) - x \cdot x}{\color{blue}{\left(0 - x\right) + z \cdot \left(z \cdot y\right)}} \]

      16. neg-sub0 1.1%

          \[\leadsto \frac{{z}^{4} \cdot \left(y \cdot y\right) - x \cdot x}{\color{blue}{\left(-x\right)} + z \cdot \left(z \cdot y\right)} \]

      17. +-commutative 1.1%

          \[\leadsto \frac{{z}^{4} \cdot \left(y \cdot y\right) - x \cdot x}{\color{blue}{z \cdot \left(z \cdot y\right) + \left(-x\right)}} \]

      18. sub-neg 1.1%

          \[\leadsto \frac{{z}^{4} \cdot \left(y \cdot y\right) - x \cdot x}{\color{blue}{z \cdot \left(z \cdot y\right) - x}} \]

      19. *-commutative 1.1%

          \[\leadsto \frac{{z}^{4} \cdot \left(y \cdot y\right) - x \cdot x}{z \cdot \color{blue}{\left(y \cdot z\right)} - x} \]

   9. Simplified 1.1%

      \[\leadsto \color{blue}{\frac{{z}^{4} \cdot \left(y \cdot y\right) - x \cdot x}{z \cdot \left(y \cdot z\right) - x}} \]

   10. Taylor expanded in z around inf 71.1%

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

       1. unpow2 71.1%

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

   12. Simplified 71.1%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \]

Alternative 4: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 2.4e+76) x (* z (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.4e+76) {
		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+76) 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+76) {
		tmp = x;
	} else {
		tmp = z * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.4e+76:
		tmp = x
	else:
		tmp = z * (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.4e+76)
		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+76)
		tmp = x;
	else
		tmp = z * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.4e+76], x, N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.4e76

   1. Initial program 99.9%

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

      1. associate-*l* 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in x around inf 61.5%

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

   if 2.4e76 < z

   1. Initial program 99.9%

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

      1. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

      1. add-cbrt-cube 69.2%

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

      2. pow3 69.2%

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

      3. add-sqr-sqrt 37.1%

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

      4. unpow-prod-down 37.1%

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

      5. pow-prod-up 37.1%

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

      6. *-commutative 37.1%

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

      7. sqrt-prod 37.1%

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

      8. sqrt-prod 37.1%

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

      9. add-sqr-sqrt 37.1%

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

      10. metadata-eval 37.1%

          \[\leadsto x + \sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{\color{blue}{6}}} \]

   5. Applied egg-rr 37.1%

      \[\leadsto x + \color{blue}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}}} \]
   6. Step-by-step derivation

      1. +-commutative 37.1%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} + x} \]

      2. flip-+ 0.3%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} \cdot \sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x}} \]

      3. pow2 0.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}}\right)}^{2}} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      4. pow1/3 0.3%

         \[\leadsto \frac{{\color{blue}{\left({\left({\left(z \cdot \sqrt{y}\right)}^{6}\right)}^{0.3333333333333333}\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      5. pow-pow 0.3%

         \[\leadsto \frac{{\color{blue}{\left({\left(z \cdot \sqrt{y}\right)}^{\left(6 \cdot 0.3333333333333333\right)}\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      6. metadata-eval 0.3%

         \[\leadsto \frac{{\left({\left(z \cdot \sqrt{y}\right)}^{\color{blue}{2}}\right)}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      7. pow2 0.3%

         \[\leadsto \frac{{\color{blue}{\left(\left(z \cdot \sqrt{y}\right) \cdot \left(z \cdot \sqrt{y}\right)\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      8. swap-sqr 0.3%

         \[\leadsto \frac{{\color{blue}{\left(\left(z \cdot z\right) \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      9. add-sqr-sqrt 0.3%

         \[\leadsto \frac{{\left(\left(z \cdot z\right) \cdot \color{blue}{y}\right)}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      10. associate-*r* 0.3%

          \[\leadsto \frac{{\color{blue}{\left(z \cdot \left(z \cdot y\right)\right)}}^{2} - x \cdot x}{\sqrt[3]{{\left(z \cdot \sqrt{y}\right)}^{6}} - x} \]

      11. pow1/3 0.3%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{{\left({\left(z \cdot \sqrt{y}\right)}^{6}\right)}^{0.3333333333333333}} - x} \]

      12. pow-pow 1.1%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{{\left(z \cdot \sqrt{y}\right)}^{\left(6 \cdot 0.3333333333333333\right)}} - x} \]

      13. metadata-eval 1.1%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{{\left(z \cdot \sqrt{y}\right)}^{\color{blue}{2}} - x} \]

      14. pow2 1.1%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{\left(z \cdot \sqrt{y}\right) \cdot \left(z \cdot \sqrt{y}\right)} - x} \]

      15. swap-sqr 0.7%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{\left(z \cdot z\right) \cdot \left(\sqrt{y} \cdot \sqrt{y}\right)} - x} \]

      16. add-sqr-sqrt 12.0%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\left(z \cdot z\right) \cdot \color{blue}{y} - x} \]

      17. associate-*r* 15.8%

          \[\leadsto \frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{\color{blue}{z \cdot \left(z \cdot y\right)} - x} \]

   7. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{\frac{{\left(z \cdot \left(z \cdot y\right)\right)}^{2} - x \cdot x}{z \cdot \left(z \cdot y\right) - x}} \]
   8. Step-by-step derivation

      1. unpow2 15.8%

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

      2. associate-*r* 13.0%

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

      3. associate-*r* 10.3%

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

   9. Applied egg-rr 10.3%

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

   10. Taylor expanded in z around inf 71.1%

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

       1. unpow2 71.1%

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

       2. associate-*r* 83.6%

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

       3. *-commutative 83.6%

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

       4. associate-*r* 83.6%

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

   12. Simplified 83.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{+76}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]

Alternative 5: 51.2% 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* 93.6%

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

3. Simplified 93.6%

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

4. Taylor expanded in x around inf 55.3%

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

5. Final simplification 55.3%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))