Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 4

Speedup: 1.2×

• 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, 1.2× 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: 87.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 10^{-82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))))
   (if (<= t_0 -1e+17) t_0 (if (<= t_0 1e-82) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if (t_0 <= -1e+17) {
		tmp = t_0;
	} else if (t_0 <= 1e-82) {
		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 <= (-1d+17)) then
        tmp = t_0
    else if (t_0 <= 1d-82) 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 <= -1e+17) {
		tmp = t_0;
	} else if (t_0 <= 1e-82) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * z)
	tmp = 0
	if t_0 <= -1e+17:
		tmp = t_0
	elif t_0 <= 1e-82:
		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 <= -1e+17)
		tmp = t_0;
	elseif (t_0 <= 1e-82)
		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 <= -1e+17)
		tmp = t_0;
	elseif (t_0 <= 1e-82)
		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, -1e+17], t$95$0, If[LessEqual[t$95$0, 1e-82], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -1e17 or 1e-82 < (*.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. *-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 78.1

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

   5. Simplified 78.1%

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

      1. associate-*r* N/A

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

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

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

      3. *-lowering-*.f64 86.7

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

   7. Applied egg-rr 86.7%

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

   if -1e17 < (*.f64 (*.f64 y z) z) < 1e-82

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 90.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -1 \cdot 10^{+17}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 10^{-82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ t_1 := y \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10^{-82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))) (t_1 (* y (* z z))))
   (if (<= t_0 -1e+17) t_1 (if (<= t_0 1e-82) 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 <= -1e+17) {
		tmp = t_1;
	} else if (t_0 <= 1e-82) {
		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 <= (-1d+17)) then
        tmp = t_1
    else if (t_0 <= 1d-82) 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 <= -1e+17) {
		tmp = t_1;
	} else if (t_0 <= 1e-82) {
		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 <= -1e+17:
		tmp = t_1
	elif t_0 <= 1e-82:
		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 <= -1e+17)
		tmp = t_1;
	elseif (t_0 <= 1e-82)
		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 <= -1e+17)
		tmp = t_1;
	elseif (t_0 <= 1e-82)
		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, -1e+17], t$95$1, If[LessEqual[t$95$0, 1e-82], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -1e17 or 1e-82 < (*.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. *-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 78.1

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

   5. Simplified 78.1%

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

   if -1e17 < (*.f64 (*.f64 y z) z) < 1e-82

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 90.3%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -1 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 10^{-82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.1% 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 52.1%

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

Reproduce

herbie shell --seed 2024204 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))