Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 6

Speedup: 1.0×

• 25.1% 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: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{\frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))))
   (if (<= t_0 -0.002) t_0 (if (<= t_0 2e-13) x (* z (/ z (/ 1.0 y)))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = t_0;
	} else if (t_0 <= 2e-13) {
		tmp = x;
	} else {
		tmp = z * (z / (1.0 / y));
	}
	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 <= (-0.002d0)) then
        tmp = t_0
    else if (t_0 <= 2d-13) then
        tmp = x
    else
        tmp = z * (z / (1.0d0 / y))
    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 <= -0.002) {
		tmp = t_0;
	} else if (t_0 <= 2e-13) {
		tmp = x;
	} else {
		tmp = z * (z / (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * z)
	tmp = 0
	if t_0 <= -0.002:
		tmp = t_0
	elif t_0 <= 2e-13:
		tmp = x
	else:
		tmp = z * (z / (1.0 / y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y * z))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = t_0;
	elseif (t_0 <= 2e-13)
		tmp = x;
	else
		tmp = Float64(z * Float64(z / Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y * z);
	tmp = 0.0;
	if (t_0 <= -0.002)
		tmp = t_0;
	elseif (t_0 <= 2e-13)
		tmp = x;
	else
		tmp = z * (z / (1.0 / y));
	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, -0.002], t$95$0, If[LessEqual[t$95$0, 2e-13], x, N[(z * N[(z / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y z) z) < -2e-3

   1. Initial program 99.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. *-lowering-*.f64 88.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 88.4%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 83.5%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 83.5%

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

      1. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{z}\right) \]

      3. *-lowering-*.f64 91.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), z\right) \]

   9. Applied egg-rr 91.5%

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

   if -2e-3 < (*.f64 (*.f64 y z) z) < 2.0000000000000001e-13

   1. Initial program 99.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. *-lowering-*.f64 97.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 97.5%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 91.6%

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

   if 2.0000000000000001e-13 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.8%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. *-lowering-*.f64 89.1%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 83.6%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 83.6%

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

      1. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{z}\right) \]

      3. *-lowering-*.f64 87.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), z\right) \]

   9. Applied egg-rr 87.7%

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

       1. associate-*r* N/A

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

       2. remove-double-neg N/A

          \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(y \cdot \left(z \cdot z\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\left(\mathsf{neg}\left(y \cdot \left(z \cdot z\right)\right)\right)\right) \]

       4. remove-double-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{1}{\mathsf{neg}\left(y \cdot \left(z \cdot z\right)\right)}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(y \cdot \left(z \cdot z\right)\right)}}\right)\right) \]

       6. frac-2neg N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{-1}{y \cdot \left(z \cdot z\right)}}\right)\right) \]

       7. associate-/r* N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{\frac{-1}{y}}{z \cdot z}}\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{z \cdot z}{\frac{-1}{y}}\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(z \cdot z\right), \left(\frac{-1}{y}\right)\right)\right) \]

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

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\frac{-1}{y}\right)\right)\right) \]

       11. /-lowering-/.f64 83.6%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{/.f64}\left(-1, y\right)\right)\right) \]

   11. Applied egg-rr 83.6%

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

       1. distribute-neg-frac N/A

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

       2. div-inv N/A

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

       3. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(z \cdot z\right)}{\mathsf{neg}\left(\frac{1}{y}\right)} \]

       4. frac-2neg N/A

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

       5. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{\frac{1}{y}}\right)}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \left(\frac{\mathsf{neg}\left(-1\right)}{y}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(-1\right)\right), \color{blue}{y}\right)\right)\right) \]

       10. metadata-eval 87.7%

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(1, y\right)\right)\right) \]

   13. Applied egg-rr 87.7%

       \[\leadsto \color{blue}{z \cdot \frac{z}{\frac{1}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -0.002:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{\frac{1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))))
   (if (<= t_0 -0.002) t_0 (if (<= t_0 2e-13) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = t_0;
	} else if (t_0 <= 2e-13) {
		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 <= (-0.002d0)) then
        tmp = t_0
    else if (t_0 <= 2d-13) 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 <= -0.002) {
		tmp = t_0;
	} else if (t_0 <= 2e-13) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * z)
	tmp = 0
	if t_0 <= -0.002:
		tmp = t_0
	elif t_0 <= 2e-13:
		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 <= -0.002)
		tmp = t_0;
	elseif (t_0 <= 2e-13)
		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 <= -0.002)
		tmp = t_0;
	elseif (t_0 <= 2e-13)
		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, -0.002], t$95$0, If[LessEqual[t$95$0, 2e-13], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -2e-3 or 2.0000000000000001e-13 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. *-lowering-*.f64 88.8%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 88.8%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 83.5%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 83.5%

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

      1. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{z}\right) \]

      3. *-lowering-*.f64 89.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), z\right) \]

   9. Applied egg-rr 89.4%

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

   if -2e-3 < (*.f64 (*.f64 y z) z) < 2.0000000000000001e-13

   1. Initial program 99.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. *-lowering-*.f64 97.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 97.5%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 91.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -0.002:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 2 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;x + y \cdot \left(z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{z}{\frac{1}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.2e+133) (+ x (* y (* z z))) (* z (/ z (/ 1.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.2e+133) {
		tmp = x + (y * (z * z));
	} else {
		tmp = z * (z / (1.0 / y));
	}
	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 <= 1.2d+133) then
        tmp = x + (y * (z * z))
    else
        tmp = z * (z / (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.2e+133) {
		tmp = x + (y * (z * z));
	} else {
		tmp = z * (z / (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.2e+133:
		tmp = x + (y * (z * z))
	else:
		tmp = z * (z / (1.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.2e+133)
		tmp = Float64(x + Float64(y * Float64(z * z)));
	else
		tmp = Float64(z * Float64(z / Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.2e+133)
		tmp = x + (y * (z * z));
	else
		tmp = z * (z / (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.2e+133], N[(x + N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.1999999999999999e133

   1. Initial program 99.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. *-lowering-*.f64 95.5%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 95.5%

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

   if 1.1999999999999999e133 < z

   1. Initial program 99.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 80.2%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 80.2%

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

      1. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot z\right), \color{blue}{z}\right) \]

      3. *-lowering-*.f64 92.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), z\right) \]

   9. Applied egg-rr 92.4%

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

       1. associate-*r* N/A

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

       2. remove-double-neg N/A

          \[\leadsto \mathsf{neg}\left(\left(\mathsf{neg}\left(y \cdot \left(z \cdot z\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\left(\mathsf{neg}\left(y \cdot \left(z \cdot z\right)\right)\right)\right) \]

       4. remove-double-div N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{1}{\mathsf{neg}\left(y \cdot \left(z \cdot z\right)\right)}}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(y \cdot \left(z \cdot z\right)\right)}}\right)\right) \]

       6. frac-2neg N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{-1}{y \cdot \left(z \cdot z\right)}}\right)\right) \]

       7. associate-/r* N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{\frac{-1}{y}}{z \cdot z}}\right)\right) \]

       8. clear-num N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{z \cdot z}{\frac{-1}{y}}\right)\right) \]

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

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(z \cdot z\right), \left(\frac{-1}{y}\right)\right)\right) \]

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

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\frac{-1}{y}\right)\right)\right) \]

       11. /-lowering-/.f64 80.2%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{/.f64}\left(-1, y\right)\right)\right) \]

   11. Applied egg-rr 80.2%

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

       1. distribute-neg-frac N/A

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

       2. div-inv N/A

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

       3. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(z \cdot z\right)}{\mathsf{neg}\left(\frac{1}{y}\right)} \]

       4. frac-2neg N/A

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

       5. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{z}{\frac{1}{y}}\right)}\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{1}{y}\right)}\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \left(\frac{\mathsf{neg}\left(-1\right)}{y}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(-1\right)\right), \color{blue}{y}\right)\right)\right) \]

       10. metadata-eval 92.4%

           \[\leadsto \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(1, y\right)\right)\right) \]

   13. Applied egg-rr 92.4%

       \[\leadsto \color{blue}{z \cdot \frac{z}{\frac{1}{y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 65.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7900000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= z 7900000.0) x (* y (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 7900000.0) {
		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 <= 7900000.0d0) 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 <= 7900000.0) {
		tmp = x;
	} else {
		tmp = y * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 7900000.0:
		tmp = x
	else:
		tmp = y * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 7900000.0)
		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 <= 7900000.0)
		tmp = x;
	else
		tmp = y * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 7900000.0], x, N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.9e6

   1. Initial program 99.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. *-lowering-*.f64 94.9%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 94.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 62.9%

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

   if 7.9e6 < z

   1. Initial program 99.9%

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

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

         \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      4. *-lowering-*.f64 88.2%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   3. Simplified 88.2%

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

   5. Taylor expanded in x around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 74.6%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 74.6%

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

Alternative 6: 51.6% 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. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(\left(y \cdot z\right) \cdot z\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(x, \left(y \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   4. *-lowering-*.f64 93.2%

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

3. Simplified 93.2%

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

5. Taylor expanded in x around inf

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

7. Simplified 51.8%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))