Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 4

Speedup: 1.0×

• 24.6% 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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 2: 66.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{-89} \lor \neg \left(z \leq 9.5 \cdot 10^{-55}\right) \land z \leq 1250000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 6.8e-89) (and (not (<= z 9.5e-55)) (<= z 1250000.0)))
   x
   (* z (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 6.8e-89) || (!(z <= 9.5e-55) && (z <= 1250000.0))) {
		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 <= 6.8d-89) .or. (.not. (z <= 9.5d-55)) .and. (z <= 1250000.0d0)) 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 <= 6.8e-89) || (!(z <= 9.5e-55) && (z <= 1250000.0))) {
		tmp = x;
	} else {
		tmp = z * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 6.8e-89) or (not (z <= 9.5e-55) and (z <= 1250000.0)):
		tmp = x
	else:
		tmp = z * (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 6.8e-89) || (!(z <= 9.5e-55) && (z <= 1250000.0)))
		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 <= 6.8e-89) || (~((z <= 9.5e-55)) && (z <= 1250000.0)))
		tmp = x;
	else
		tmp = z * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 6.8e-89], And[N[Not[LessEqual[z, 9.5e-55]], $MachinePrecision], LessEqual[z, 1250000.0]]], x, N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.8000000000000001e-89 or 9.5000000000000006e-55 < z < 1.25e6

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 61.5%

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

   if 6.8000000000000001e-89 < z < 9.5000000000000006e-55 or 1.25e6 < z

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      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* 99.6%

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

      4. fma-define 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in y around inf 74.8%

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

      1. metadata-eval 74.8%

         \[\leadsto y \cdot {z}^{\color{blue}{\left(\frac{4}{2}\right)}} \]

      2. sqrt-pow2 74.5%

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

      3. add-sqr-sqrt 45.6%

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

      4. sqrt-unprod 37.4%

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

      5. sqrt-pow2 37.5%

         \[\leadsto \sqrt{\left(y \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}}\right) \cdot \left(y \cdot {\left(\sqrt{z}\right)}^{4}\right)} \]

      6. metadata-eval 37.5%

         \[\leadsto \sqrt{\left(y \cdot {z}^{\color{blue}{2}}\right) \cdot \left(y \cdot {\left(\sqrt{z}\right)}^{4}\right)} \]

      7. sqrt-pow2 37.5%

         \[\leadsto \sqrt{\left(y \cdot {z}^{2}\right) \cdot \left(y \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}}\right)} \]

      8. metadata-eval 37.5%

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

      9. *-commutative 37.5%

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

      10. *-commutative 37.5%

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

      11. swap-sqr 30.6%

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

      12. pow-sqr 30.6%

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

      13. metadata-eval 30.6%

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

      14. pow2 30.6%

          \[\leadsto \sqrt{{z}^{4} \cdot \color{blue}{{y}^{2}}} \]

   7. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot {y}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 30.6%

         \[\leadsto \sqrt{\color{blue}{{y}^{2} \cdot {z}^{4}}} \]

      2. sqrt-prod 30.6%

         \[\leadsto \color{blue}{\sqrt{{y}^{2}} \cdot \sqrt{{z}^{4}}} \]

      3. unpow2 30.6%

         \[\leadsto \sqrt{\color{blue}{y \cdot y}} \cdot \sqrt{{z}^{4}} \]

      4. sqrt-prod 36.6%

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

      5. sqrt-pow1 45.7%

         \[\leadsto \left(\sqrt{y} \cdot \sqrt{y}\right) \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}} \]

      6. metadata-eval 45.7%

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

      7. pow2 45.7%

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

      8. add-sqr-sqrt 74.8%

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

      9. associate-*r* 82.1%

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

      10. *-commutative 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{-89} \lor \neg \left(z \leq 9.5 \cdot 10^{-55}\right) \land z \leq 1250000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.35e+154) (+ x (* y (* z z))) (* z (* y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.35e+154) {
		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 <= 1.35d+154) 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 <= 1.35e+154) {
		tmp = x + (y * (z * z));
	} else {
		tmp = z * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.35e+154:
		tmp = x + (y * (z * z))
	else:
		tmp = z * (y * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.35e+154)
		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 <= 1.35e+154)
		tmp = x + (y * (z * z));
	else
		tmp = z * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.35e+154], 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 < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. add-sqr-sqrt 64.4%

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

      2. pow2 64.4%

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

      3. associate-*l* 67.2%

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

      4. sqrt-prod 48.5%

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

      5. sqrt-prod 19.4%

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

      6. add-sqr-sqrt 51.0%

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

   4. Applied egg-rr 51.0%

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

      1. unpow2 51.0%

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

      2. swap-sqr 48.5%

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

      3. add-sqr-sqrt 95.4%

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

      4. associate-*r* 99.9%

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

      5. add-sqr-sqrt 42.7%

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

      6. associate-*l* 42.7%

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

      7. *-commutative 42.7%

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

      8. associate-*l* 42.8%

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

      9. pow1 42.8%

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

      10. pow1/2 42.8%

          \[\leadsto x + \sqrt{z} \cdot \left(y \cdot \left({z}^{1} \cdot \color{blue}{{z}^{0.5}}\right)\right) \]

      11. pow-prod-up 42.8%

          \[\leadsto x + \sqrt{z} \cdot \left(y \cdot \color{blue}{{z}^{\left(1 + 0.5\right)}}\right) \]

      12. metadata-eval 42.8%

          \[\leadsto x + \sqrt{z} \cdot \left(y \cdot {z}^{\color{blue}{1.5}}\right) \]

   6. Applied egg-rr 42.8%

      \[\leadsto x + \color{blue}{\sqrt{z} \cdot \left(y \cdot {z}^{1.5}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 42.8%

         \[\leadsto x + \color{blue}{\left(y \cdot {z}^{1.5}\right) \cdot \sqrt{z}} \]

      2. associate-*l* 42.0%

         \[\leadsto x + \color{blue}{y \cdot \left({z}^{1.5} \cdot \sqrt{z}\right)} \]

      3. unpow1/2 42.0%

         \[\leadsto x + y \cdot \left({z}^{1.5} \cdot \color{blue}{{z}^{0.5}}\right) \]

      4. metadata-eval 42.0%

         \[\leadsto x + y \cdot \left({z}^{1.5} \cdot {z}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right) \]

      5. pow-sqr 42.0%

         \[\leadsto x + y \cdot \left({z}^{1.5} \cdot \color{blue}{\left({z}^{0.25} \cdot {z}^{0.25}\right)}\right) \]

      6. metadata-eval 42.0%

         \[\leadsto x + y \cdot \left({z}^{\color{blue}{\left(0.5 + 1\right)}} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      7. pow-plus 42.0%

         \[\leadsto x + y \cdot \left(\color{blue}{\left({z}^{0.5} \cdot z\right)} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      8. unpow1/2 42.0%

         \[\leadsto x + y \cdot \left(\left(\color{blue}{\sqrt{z}} \cdot z\right) \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      9. rem-square-sqrt 41.9%

         \[\leadsto x + y \cdot \left(\left(\sqrt{z} \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}\right) \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      10. cube-mult 41.9%

          \[\leadsto x + y \cdot \left(\color{blue}{{\left(\sqrt{z}\right)}^{3}} \cdot \left({z}^{0.25} \cdot {z}^{0.25}\right)\right) \]

      11. pow-sqr 41.9%

          \[\leadsto x + y \cdot \left({\left(\sqrt{z}\right)}^{3} \cdot \color{blue}{{z}^{\left(2 \cdot 0.25\right)}}\right) \]

      12. metadata-eval 41.9%

          \[\leadsto x + y \cdot \left({\left(\sqrt{z}\right)}^{3} \cdot {z}^{\color{blue}{0.5}}\right) \]

      13. unpow1/2 41.9%

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

      14. pow-plus 41.9%

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

      15. metadata-eval 41.9%

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

   8. Simplified 41.9%

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

      1. sqrt-pow2 95.4%

         \[\leadsto x + y \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}} \]

      2. metadata-eval 95.4%

         \[\leadsto x + y \cdot {z}^{\color{blue}{2}} \]

      3. unpow2 95.4%

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

   10. Applied egg-rr 95.4%

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

   if 1.35000000000000003e154 < 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 99.9%

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

      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 77.3%

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

      1. metadata-eval 77.3%

         \[\leadsto y \cdot {z}^{\color{blue}{\left(\frac{4}{2}\right)}} \]

      2. sqrt-pow2 77.3%

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

      3. add-sqr-sqrt 48.6%

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

      4. sqrt-unprod 48.9%

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

      5. sqrt-pow2 48.9%

         \[\leadsto \sqrt{\left(y \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}}\right) \cdot \left(y \cdot {\left(\sqrt{z}\right)}^{4}\right)} \]

      6. metadata-eval 48.9%

         \[\leadsto \sqrt{\left(y \cdot {z}^{\color{blue}{2}}\right) \cdot \left(y \cdot {\left(\sqrt{z}\right)}^{4}\right)} \]

      7. sqrt-pow2 48.9%

         \[\leadsto \sqrt{\left(y \cdot {z}^{2}\right) \cdot \left(y \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}}\right)} \]

      8. metadata-eval 48.9%

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

      9. *-commutative 48.9%

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

      10. *-commutative 48.9%

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

      11. swap-sqr 44.0%

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

      12. pow-sqr 44.0%

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

      13. metadata-eval 44.0%

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

      14. pow2 44.0%

          \[\leadsto \sqrt{{z}^{4} \cdot \color{blue}{{y}^{2}}} \]

   7. Applied egg-rr 44.0%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot {y}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 44.0%

         \[\leadsto \sqrt{\color{blue}{{y}^{2} \cdot {z}^{4}}} \]

      2. sqrt-prod 44.0%

         \[\leadsto \color{blue}{\sqrt{{y}^{2}} \cdot \sqrt{{z}^{4}}} \]

      3. unpow2 44.0%

         \[\leadsto \sqrt{\color{blue}{y \cdot y}} \cdot \sqrt{{z}^{4}} \]

      4. sqrt-prod 48.6%

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

      5. sqrt-pow1 48.6%

         \[\leadsto \left(\sqrt{y} \cdot \sqrt{y}\right) \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}} \]

      6. metadata-eval 48.6%

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

      7. pow2 48.6%

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

      8. add-sqr-sqrt 77.3%

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

      9. associate-*r* 95.9%

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

      10. *-commutative 95.9%

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

   9. Applied egg-rr 95.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.7% 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 50.7%

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

4. Final simplification 50.7%

   \[\leadsto x \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))