Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 9.6s

Alternatives: 4

Speedup: 1.0×

• 25.3% 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: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+85} \lor \neg \left(t\_0 \leq 10^{-95}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))))
   (if (or (<= t_0 -2e+85) (not (<= t_0 1e-95))) t_0 x)))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if ((t_0 <= -2e+85) || !(t_0 <= 1e-95)) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	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 <= (-2d+85)) .or. (.not. (t_0 <= 1d-95))) then
        tmp = t_0
    else
        tmp = x
    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 <= -2e+85) || !(t_0 <= 1e-95)) {
		tmp = t_0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * z)
	tmp = 0
	if (t_0 <= -2e+85) or not (t_0 <= 1e-95):
		tmp = t_0
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(y * z))
	tmp = 0.0
	if ((t_0 <= -2e+85) || !(t_0 <= 1e-95))
		tmp = t_0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (y * z);
	tmp = 0.0;
	if ((t_0 <= -2e+85) || ~((t_0 <= 1e-95)))
		tmp = t_0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e+85], N[Not[LessEqual[t$95$0, 1e-95]], $MachinePrecision]], t$95$0, x]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -2e85 or 9.99999999999999989e-96 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 84.3%

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

   4. Taylor expanded in y around inf 73.1%

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

      1. associate-*r/ 71.5%

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

   6. Simplified 71.5%

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

      1. associate-*r* 76.3%

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

      2. associate-*r/ 76.0%

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

      3. *-commutative 76.0%

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

      4. *-commutative 76.0%

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

   8. Applied egg-rr 76.0%

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

      1. *-commutative 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*r/ 76.3%

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

      4. unpow2 76.3%

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

      5. associate-*r/ 77.0%

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

      6. *-commutative 77.0%

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

      7. associate-*r* 79.3%

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

      8. *-commutative 79.3%

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

   10. Applied egg-rr 79.3%

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

   11. Taylor expanded in y around 0 87.8%

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

   if -2e85 < (*.f64 (*.f64 y z) z) < 9.99999999999999989e-96

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 80.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -2 \cdot 10^{+85} \lor \neg \left(z \cdot \left(y \cdot z\right) \leq 10^{-95}\right):\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-77} \lor \neg \left(z \leq 3.5 \cdot 10^{+15}\right) \land z \leq 1.48 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 2.1e-77) (and (not (<= z 3.5e+15)) (<= z 1.48e+16)))
   x
   (* y (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 2.1e-77) || (!(z <= 3.5e+15) && (z <= 1.48e+16))) {
		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 <= 2.1d-77) .or. (.not. (z <= 3.5d+15)) .and. (z <= 1.48d+16)) 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 <= 2.1e-77) || (!(z <= 3.5e+15) && (z <= 1.48e+16))) {
		tmp = x;
	} else {
		tmp = y * (z * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 2.1e-77) or (not (z <= 3.5e+15) and (z <= 1.48e+16)):
		tmp = x
	else:
		tmp = y * (z * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 2.1e-77) || (!(z <= 3.5e+15) && (z <= 1.48e+16)))
		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 <= 2.1e-77) || (~((z <= 3.5e+15)) && (z <= 1.48e+16)))
		tmp = x;
	else
		tmp = y * (z * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 2.1e-77], And[N[Not[LessEqual[z, 3.5e+15]], $MachinePrecision], LessEqual[z, 1.48e+16]]], x, N[(y * N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.10000000000000015e-77 or 3.5e15 < z < 1.48e16

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 57.9%

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

   if 2.10000000000000015e-77 < z < 3.5e15 or 1.48e16 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf 92.2%

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

   4. Taylor expanded in y around inf 70.0%

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

      1. associate-*r/ 64.8%

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

   6. Simplified 64.8%

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

      1. unpow2 64.8%

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

      2. associate-/l* 64.8%

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

   8. Applied egg-rr 64.8%

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

      1. associate-*r* 65.4%

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

      2. associate-*r* 66.7%

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

      3. clear-num 66.7%

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

      4. un-div-inv 66.7%

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

      5. *-commutative 66.7%

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

      6. *-commutative 66.7%

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

   10. Applied egg-rr 66.7%

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

       1. associate-/r/ 66.9%

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

       2. *-commutative 66.9%

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

       3. associate-*r/ 66.8%

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

       4. associate-*l* 71.1%

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

       5. associate-*l* 70.0%

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

       6. associate-*r/ 73.9%

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

   12. Applied egg-rr 73.9%

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

   13. Taylor expanded in x around 0 73.9%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-77} \lor \neg \left(z \leq 3.5 \cdot 10^{+15}\right) \land z \leq 1.48 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.6% 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. Add Preprocessing

3. Taylor expanded in x around inf 48.0%

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

Reproduce

herbie shell --seed 2024107 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))