Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 8.7s

Alternatives: 4

Speedup: 1.2×

• 25.2% 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: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))))
   (if (<= t_0 -0.01) t_0 (if (<= t_0 5e-30) (* 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 <= -0.01) {
		tmp = t_0;
	} else if (t_0 <= 5e-30) {
		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 <= (-0.01d0)) then
        tmp = t_0
    else if (t_0 <= 5d-30) 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 <= -0.01) {
		tmp = t_0;
	} else if (t_0 <= 5e-30) {
		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 <= -0.01:
		tmp = t_0
	elif t_0 <= 5e-30:
		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 <= -0.01)
		tmp = t_0;
	elseif (t_0 <= 5e-30)
		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 <= -0.01)
		tmp = t_0;
	elseif (t_0 <= 5e-30)
		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, -0.01], t$95$0, If[LessEqual[t$95$0, 5e-30], N[(x * 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -0.0100000000000000002 or 4.99999999999999972e-30 < (*.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 80.0

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

   5. Applied rewrites 80.0%

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

   7. Applied rewrites 89.9%

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

   if -0.0100000000000000002 < (*.f64 (*.f64 y z) z) < 4.99999999999999972e-30

   1. Initial program 100.0%

      \[x + \left(y \cdot z\right) \cdot z \]
   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. unpow1 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 71.3

          \[\leadsto \mathsf{fma}\left({\left(y \cdot \left(z \cdot z\right)\right)}^{0.5}, {\left(y \cdot \left(z \cdot z\right)\right)}^{\color{blue}{0.5}}, x\right) \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(y \cdot \left(z \cdot z\right)\right)}^{0.5}, {\left(y \cdot \left(z \cdot z\right)\right)}^{0.5}, x\right)} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

   9. Applied rewrites 92.0%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 88.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -0.01:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 5 \cdot 10^{-30}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ t_1 := y \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-30}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))) (t_1 (* y (* z z))))
   (if (<= t_0 -0.01) t_1 (if (<= t_0 5e-30) (* 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 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 5e-30) {
		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 <= (-0.01d0)) then
        tmp = t_1
    else if (t_0 <= 5d-30) 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 <= -0.01) {
		tmp = t_1;
	} else if (t_0 <= 5e-30) {
		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 <= -0.01:
		tmp = t_1
	elif t_0 <= 5e-30:
		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 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 5e-30)
		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 <= -0.01)
		tmp = t_1;
	elseif (t_0 <= 5e-30)
		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, -0.01], t$95$1, If[LessEqual[t$95$0, 5e-30], N[(x * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -0.0100000000000000002 or 4.99999999999999972e-30 < (*.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 80.0

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

   5. Applied rewrites 80.0%

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

   if -0.0100000000000000002 < (*.f64 (*.f64 y z) z) < 4.99999999999999972e-30

   1. Initial program 100.0%

      \[x + \left(y \cdot z\right) \cdot z \]
   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. unpow1 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. metadata-eval 71.3

          \[\leadsto \mathsf{fma}\left({\left(y \cdot \left(z \cdot z\right)\right)}^{0.5}, {\left(y \cdot \left(z \cdot z\right)\right)}^{\color{blue}{0.5}}, x\right) \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(y \cdot \left(z \cdot z\right)\right)}^{0.5}, {\left(y \cdot \left(z \cdot z\right)\right)}^{0.5}, x\right)} \]

   5. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 84.5

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

   7. Applied rewrites 84.5%

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

   9. Applied rewrites 92.0%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 88.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -0.01:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 5 \cdot 10^{-30}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.4% 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
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. unpow1 N/A

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

   4. sqr-pow N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. lower-pow.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 59.7

       \[\leadsto \mathsf{fma}\left({\left(y \cdot \left(z \cdot z\right)\right)}^{0.5}, {\left(y \cdot \left(z \cdot z\right)\right)}^{\color{blue}{0.5}}, x\right) \]

4. Applied rewrites 59.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(y \cdot \left(z \cdot z\right)\right)}^{0.5}, {\left(y \cdot \left(z \cdot z\right)\right)}^{0.5}, x\right)} \]

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 80.0

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

7. Applied rewrites 80.0%

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

9. Applied rewrites 84.7%

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

10. Taylor expanded in z around 0

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

12. Applied rewrites 48.4%

    \[\leadsto 1 \cdot x \]

13. Final simplification 48.4%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))