Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 12.5s

Alternatives: 4

Speedup: 1.2×

• 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, 1.2× 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. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -200000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-112}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))))
   (if (<= t_0 -200000000.0) t_0 (if (<= t_0 5e-112) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_0;
	} else if (t_0 <= 5e-112) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = z * (y * z)
    if (t_0 <= (-200000000.0d0)) then
        tmp = t_0
    else if (t_0 <= 5d-112) then
        tmp = x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_0;
	} else if (t_0 <= 5e-112) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * z)
	tmp = 0
	if t_0 <= -200000000.0:
		tmp = t_0
	elif t_0 <= 5e-112:
		tmp = x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y * z))
	tmp = 0.0
	if (t_0 <= -200000000.0)
		tmp = t_0;
	elseif (t_0 <= 5e-112)
		tmp = x;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y * z);
	tmp = 0.0;
	if (t_0 <= -200000000.0)
		tmp = t_0;
	elseif (t_0 <= 5e-112)
		tmp = x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000000.0], t$95$0, If[LessEqual[t$95$0, 5e-112], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -2e8 or 5.00000000000000044e-112 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(y, z \cdot z, x\right)}}} \]

      10. *-lowering-*.f64 90.1

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(y, \color{blue}{z \cdot z}, x\right)}} \]

   4. Applied egg-rr 90.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y, z \cdot z, x\right)}}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot \frac{x}{y \cdot {z}^{4}} + \frac{1}{{z}^{2}}}{y}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{{z}^{2}} + -1 \cdot \frac{x}{y \cdot {z}^{4}}}}{y}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{\frac{1}{{z}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y \cdot {z}^{4}}\right)\right)}}{y}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{{z}^{2}} - \frac{x}{y \cdot {z}^{4}}}}{y}} \]

      4. div-sub N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{{z}^{2}}}{y} - \frac{\frac{x}{y \cdot {z}^{4}}}{y}}} \]

      5. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{y \cdot {z}^{2}}} - \frac{\frac{x}{y \cdot {z}^{4}}}{y}} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{1}{\frac{1}{y \cdot {z}^{2}} - \color{blue}{\frac{x}{y \cdot \left(y \cdot {z}^{4}\right)}}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{1}{y \cdot {z}^{2}} - \frac{x}{\color{blue}{\left(y \cdot y\right) \cdot {z}^{4}}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{\frac{1}{y \cdot {z}^{2}} - \frac{x}{\color{blue}{{y}^{2}} \cdot {z}^{4}}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{y}}{{z}^{2}}} - \frac{x}{{y}^{2} \cdot {z}^{4}}} \]

      10. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \color{blue}{\frac{\frac{x}{{y}^{2}}}{{z}^{4}}}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \frac{\frac{x}{{y}^{2}}}{{z}^{\color{blue}{\left(2 \cdot 2\right)}}}} \]

      12. pow-sqr N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \frac{\frac{x}{{y}^{2}}}{\color{blue}{{z}^{2} \cdot {z}^{2}}}} \]

      13. associate-/l/ N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \color{blue}{\frac{\frac{\frac{x}{{y}^{2}}}{{z}^{2}}}{{z}^{2}}}} \]

      14. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \frac{\color{blue}{\frac{x}{{y}^{2} \cdot {z}^{2}}}}{{z}^{2}}} \]

      15. div-sub N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{y} - \frac{x}{{y}^{2} \cdot {z}^{2}}}{{z}^{2}}}} \]

   7. Simplified 75.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{y} - \frac{x}{y \cdot \mathsf{fma}\left(y, z \cdot z, 0\right)}}{z \cdot z}}} \]

   8. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{y \cdot {z}^{2}}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{y \cdot {z}^{2}}}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{y \cdot {z}^{2}}}} \]

      3. unpow2 N/A

         \[\leadsto \frac{1}{\frac{1}{y \cdot \color{blue}{\left(z \cdot z\right)}}} \]

      4. *-lowering-*.f64 76.7

         \[\leadsto \frac{1}{\frac{1}{y \cdot \color{blue}{\left(z \cdot z\right)}}} \]

   10. Simplified 76.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{1}{y \cdot \left(z \cdot z\right)}}} \]
   11. Step-by-step derivation

       1. remove-double-div N/A

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

       2. associate-*r* N/A

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

       3. *-lowering-*.f64 N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 86.4

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

   12. Applied egg-rr 86.4%

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

   if -2e8 < (*.f64 (*.f64 y z) z) < 5.00000000000000044e-112

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 91.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -200000000:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 5 \cdot 10^{-112}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ t_1 := y \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -200000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))) (t_1 (* y (* z z))))
   (if (<= t_0 -200000000.0) t_1 (if (<= t_0 5e+77) x t_1))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double t_1 = y * (z * z);
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+77) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * (y * z)
    t_1 = y * (z * z)
    if (t_0 <= (-200000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 5d+77) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double t_1 = y * (z * z);
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+77) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * z)
	t_1 = y * (z * z)
	tmp = 0
	if t_0 <= -200000000.0:
		tmp = t_1
	elif t_0 <= 5e+77:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y * z))
	t_1 = Float64(y * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -200000000.0)
		tmp = t_1;
	elseif (t_0 <= 5e+77)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y * z);
	t_1 = y * (z * z);
	tmp = 0.0;
	if (t_0 <= -200000000.0)
		tmp = t_1;
	elseif (t_0 <= 5e+77)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000000.0], t$95$1, If[LessEqual[t$95$0, 5e+77], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -2e8 or 5.00000000000000004e77 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

      6. /-lowering-/.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(y, z \cdot z, x\right)}}} \]

      10. *-lowering-*.f64 89.1

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(y, \color{blue}{z \cdot z}, x\right)}} \]

   4. Applied egg-rr 89.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(y, z \cdot z, x\right)}}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot \frac{x}{y \cdot {z}^{4}} + \frac{1}{{z}^{2}}}{y}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{{z}^{2}} + -1 \cdot \frac{x}{y \cdot {z}^{4}}}}{y}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{\frac{1}{{z}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{y \cdot {z}^{4}}\right)\right)}}{y}} \]

      3. unsub-neg N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{{z}^{2}} - \frac{x}{y \cdot {z}^{4}}}}{y}} \]

      4. div-sub N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{{z}^{2}}}{y} - \frac{\frac{x}{y \cdot {z}^{4}}}{y}}} \]

      5. associate-/l/ N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{y \cdot {z}^{2}}} - \frac{\frac{x}{y \cdot {z}^{4}}}{y}} \]

      6. associate-/l/ N/A

         \[\leadsto \frac{1}{\frac{1}{y \cdot {z}^{2}} - \color{blue}{\frac{x}{y \cdot \left(y \cdot {z}^{4}\right)}}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{1}{y \cdot {z}^{2}} - \frac{x}{\color{blue}{\left(y \cdot y\right) \cdot {z}^{4}}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{\frac{1}{y \cdot {z}^{2}} - \frac{x}{\color{blue}{{y}^{2}} \cdot {z}^{4}}} \]

      9. associate-/r* N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{y}}{{z}^{2}}} - \frac{x}{{y}^{2} \cdot {z}^{4}}} \]

      10. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \color{blue}{\frac{\frac{x}{{y}^{2}}}{{z}^{4}}}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \frac{\frac{x}{{y}^{2}}}{{z}^{\color{blue}{\left(2 \cdot 2\right)}}}} \]

      12. pow-sqr N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \frac{\frac{x}{{y}^{2}}}{\color{blue}{{z}^{2} \cdot {z}^{2}}}} \]

      13. associate-/l/ N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \color{blue}{\frac{\frac{\frac{x}{{y}^{2}}}{{z}^{2}}}{{z}^{2}}}} \]

      14. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{\frac{1}{y}}{{z}^{2}} - \frac{\color{blue}{\frac{x}{{y}^{2} \cdot {z}^{2}}}}{{z}^{2}}} \]

      15. div-sub N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{y} - \frac{x}{{y}^{2} \cdot {z}^{2}}}{{z}^{2}}}} \]

   7. Simplified 82.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{y} - \frac{x}{y \cdot \mathsf{fma}\left(y, z \cdot z, 0\right)}}{z \cdot z}}} \]

   8. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 82.5

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

   10. Simplified 82.5%

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

   if -2e8 < (*.f64 (*.f64 y z) z) < 5.00000000000000004e77

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 85.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -200000000:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 5 \cdot 10^{+77}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.3% accurate, 14.0× 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

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 51.6%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))