Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

Alternatives: 4

Speedup: 1.2×

• 24.8% 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

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. 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: 87.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))))
   (if (<= t_0 -2000000.0) t_0 (if (<= t_0 2e+19) x t_0))))

C

double code(double x, double y, double z) {
	double t_0 = z * (y * z);
	double tmp;
	if (t_0 <= -2000000.0) {
		tmp = t_0;
	} else if (t_0 <= 2e+19) {
		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 <= (-2000000.0d0)) then
        tmp = t_0
    else if (t_0 <= 2d+19) 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 <= -2000000.0) {
		tmp = t_0;
	} else if (t_0 <= 2e+19) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (y * z)
	tmp = 0
	if t_0 <= -2000000.0:
		tmp = t_0
	elif t_0 <= 2e+19:
		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 <= -2000000.0)
		tmp = t_0;
	elseif (t_0 <= 2e+19)
		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 <= -2000000.0)
		tmp = t_0;
	elseif (t_0 <= 2e+19)
		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, -2000000.0], t$95$0, If[LessEqual[t$95$0, 2e+19], x, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -2e6 or 2e19 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.9%

      \[x + \left(y \cdot z\right) \cdot z \]
   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 77.4

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

   5. Applied rewrites 77.4%

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

      1. associate-*r* N/A

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

      2. lift-*.f64 N/A

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

      3. lower-*.f64 91.8

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

   7. Applied rewrites 91.8%

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

   if -2e6 < (*.f64 (*.f64 y z) z) < 2e19

   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

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. 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. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 94.9

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(x\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 85.8%

       \[\leadsto \color{blue}{-1} \cdot \left(-x\right) \]
   11. Step-by-step derivation

       1. distribute-rgt-neg-out N/A

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

       2. neg-mul-1 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

       3. remove-double-neg 85.8

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

   12. Applied rewrites 85.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -2000000:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 2 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \left(y \cdot z\right)\\ t_1 := y \cdot \left(z \cdot z\right)\\ \mathbf{if}\;t\_0 \leq -2000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (* y z))) (t_1 (* y (* z z))))
   (if (<= t_0 -2000000.0) t_1 (if (<= t_0 2e+19) 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 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+19) {
		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 <= (-2000000.0d0)) then
        tmp = t_1
    else if (t_0 <= 2d+19) 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 <= -2000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e+19) {
		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 <= -2000000.0:
		tmp = t_1
	elif t_0 <= 2e+19:
		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 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e+19)
		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 <= -2000000.0)
		tmp = t_1;
	elseif (t_0 <= 2e+19)
		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, -2000000.0], t$95$1, If[LessEqual[t$95$0, 2e+19], x, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y z) z) < -2e6 or 2e19 < (*.f64 (*.f64 y z) z)

   1. Initial program 99.9%

      \[x + \left(y \cdot z\right) \cdot z \]
   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 77.4

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

   5. Applied rewrites 77.4%

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

   if -2e6 < (*.f64 (*.f64 y z) z) < 2e19

   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

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. 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. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 94.9

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(x\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 85.8%

       \[\leadsto \color{blue}{-1} \cdot \left(-x\right) \]
   11. Step-by-step derivation

       1. distribute-rgt-neg-out N/A

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

       2. neg-mul-1 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

       3. remove-double-neg 85.8

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

   12. Applied rewrites 85.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(y \cdot z\right) \leq -2000000:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;z \cdot \left(y \cdot z\right) \leq 2 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.9% 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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. 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. Taylor expanded in x around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. associate-/l* N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 84.0

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

7. Applied rewrites 84.0%

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

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(x\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 49.0%

    \[\leadsto \color{blue}{-1} \cdot \left(-x\right) \]
11. Step-by-step derivation

    1. distribute-rgt-neg-out N/A

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

    2. neg-mul-1 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

    3. remove-double-neg 49.0

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

12. Applied rewrites 49.0%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))