Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 7.7s

Alternatives: 4

Speedup: 1.2×

• 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, 1.2× 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. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))))
   (if (<= t_0 -5e-38) t_0 (if (<= t_0 2e+96) (* x 1.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if (t_0 <= -5e-38) {
		tmp = t_0;
	} else if (t_0 <= 2e+96) {
		tmp = x * 1.0;
	} 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 <= (-5d-38)) then
        tmp = t_0
    else if (t_0 <= 2d+96) then
        tmp = x * 1.0d0
    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 <= -5e-38) {
		tmp = t_0;
	} else if (t_0 <= 2e+96) {
		tmp = x * 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * z)
	tmp = 0
	if t_0 <= -5e-38:
		tmp = t_0
	elif t_0 <= 2e+96:
		tmp = x * 1.0
	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 <= -5e-38)
		tmp = t_0;
	elseif (t_0 <= 2e+96)
		tmp = Float64(x * 1.0);
	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 <= -5e-38)
		tmp = t_0;
	elseif (t_0 <= 2e+96)
		tmp = x * 1.0;
	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, -5e-38], t$95$0, If[LessEqual[t$95$0, 2e+96], N[(x * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -5.00000000000000033e-38 or 2.0000000000000001e96 < (*.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 0

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

      1. lower-*.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. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 89.7%

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

   if -5.00000000000000033e-38 < (*.f64 (*.f64 y z) z) < 2.0000000000000001e96

   1. Initial program 100.0%

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

   4. Applied rewrites 28.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{y \cdot \left({z}^{2} \cdot \left(-1 \cdot \left({y}^{2} \cdot {z}^{4}\right) + {y}^{2} \cdot {z}^{4}\right)\right)}{{x}^{3}}\right) - \left(-1 \cdot \frac{y \cdot {z}^{2}}{x} + \left(-1 \cdot \frac{{y}^{2} \cdot {z}^{4}}{{x}^{2}} + \frac{{y}^{2} \cdot {z}^{4}}{{x}^{2}}\right)\right)\right)} \]

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 87.4%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -5 \cdot 10^{-38}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ t_1 := y \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))) (t_1 (* y (* z z))))
   (if (<= t_0 -5e-38) t_1 (if (<= t_0 2e+96) (* x 1.0) 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 <= -5e-38) {
		tmp = t_1;
	} else if (t_0 <= 2e+96) {
		tmp = x * 1.0;
	} 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 <= (-5d-38)) then
        tmp = t_1
    else if (t_0 <= 2d+96) then
        tmp = x * 1.0d0
    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 <= -5e-38) {
		tmp = t_1;
	} else if (t_0 <= 2e+96) {
		tmp = x * 1.0;
	} 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 <= -5e-38:
		tmp = t_1
	elif t_0 <= 2e+96:
		tmp = x * 1.0
	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 <= -5e-38)
		tmp = t_1;
	elseif (t_0 <= 2e+96)
		tmp = Float64(x * 1.0);
	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 <= -5e-38)
		tmp = t_1;
	elseif (t_0 <= 2e+96)
		tmp = x * 1.0;
	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, -5e-38], t$95$1, If[LessEqual[t$95$0, 2e+96], N[(x * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -5.00000000000000033e-38 or 2.0000000000000001e96 < (*.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 0

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

      1. lower-*.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. lower-*.f64 82.2

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

   5. Applied rewrites 82.2%

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

   if -5.00000000000000033e-38 < (*.f64 (*.f64 y z) z) < 2.0000000000000001e96

   1. Initial program 100.0%

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

   4. Applied rewrites 28.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{y \cdot \left({z}^{2} \cdot \left(-1 \cdot \left({y}^{2} \cdot {z}^{4}\right) + {y}^{2} \cdot {z}^{4}\right)\right)}{{x}^{3}}\right) - \left(-1 \cdot \frac{y \cdot {z}^{2}}{x} + \left(-1 \cdot \frac{{y}^{2} \cdot {z}^{4}}{{x}^{2}} + \frac{{y}^{2} \cdot {z}^{4}}{{x}^{2}}\right)\right)\right)} \]

   7. Applied rewrites 88.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 87.4%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -5 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 2 \cdot 10^{+96}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 1.0))

C

double code(double x, double y, double z) {
	return x * 1.0;
}

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 * 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return x * 1.0;
}

Python

def code(x, y, z):
	return x * 1.0

Julia

function code(x, y, z)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 19.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{y \cdot \left({z}^{2} \cdot \left(-1 \cdot \left({y}^{2} \cdot {z}^{4}\right) + {y}^{2} \cdot {z}^{4}\right)\right)}{{x}^{3}}\right) - \left(-1 \cdot \frac{y \cdot {z}^{2}}{x} + \left(-1 \cdot \frac{{y}^{2} \cdot {z}^{4}}{{x}^{2}} + \frac{{y}^{2} \cdot {z}^{4}}{{x}^{2}}\right)\right)\right)} \]

7. Applied rewrites 65.1%

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

8. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 51.4%

    \[\leadsto x \cdot 1 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024225 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))