Linear.Quaternion:$c/ from linear-1.19.1.3, A

Percentage Accurate: 98.1% → 99.3%

Time: 6.3s

Alternatives: 5

Speedup: 1.8×

• 33.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (((x * y) + (z * z)) + (z * z)) + (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)) + (z * z)) + (z * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (((x * y) + (z * z)) + (z * z)) + (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)) + (z * z)) + (z * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, \left(z \cdot z\right) \cdot 3\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma y x (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	return fma(y, x, ((z * z) * 3.0));
}

Julia

function code(x, y, z)
	return fma(y, x, Float64(Float64(z * z) * 3.0))
end

Wolfram

code[x_, y_, z_] := N[(y * x + N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

      \[\leadsto \color{blue}{3} \cdot {z}^{2} + x \cdot y \]

   5. unpow2 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} + x \cdot y \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(3 \cdot z\right) \cdot z} + x \cdot y \]

   7. remove-double-neg N/A

      \[\leadsto \left(3 \cdot z\right) \cdot z + x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \left(3 \cdot z\right) \cdot z + x \cdot \left(\mathsf{neg}\left(\color{blue}{-1 \cdot y}\right)\right) \]

   9. distribute-rgt-neg-in N/A

      \[\leadsto \left(3 \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(-1 \cdot y\right)\right)\right)} \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \left(3 \cdot z\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot z, z, \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{3 \cdot z}, z, \left(\mathsf{neg}\left(x\right)\right) \cdot \left(-1 \cdot y\right)\right) \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{\mathsf{neg}\left(x \cdot \left(-1 \cdot y\right)\right)}\right) \]

   14. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{x \cdot \left(\mathsf{neg}\left(-1 \cdot y\right)\right)}\right) \]

   15. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(3 \cdot z, z, x \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right) \]

   16. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(3 \cdot z, z, x \cdot \color{blue}{y}\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{y \cdot x}\right) \]

   18. lower-*.f64 98.8

       \[\leadsto \mathsf{fma}\left(3 \cdot z, z, \color{blue}{y \cdot x}\right) \]

5. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(3 \cdot z, z, y \cdot x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.9%

   \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, \left(z \cdot z\right) \cdot 3\right) \]
8. Add Preprocessing

Alternative 2: 80.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.5e+59) (fma z z (* y x)) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.5e+59) {
		tmp = fma(z, z, (y * x));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.5e+59)
		tmp = fma(z, z, Float64(y * x));
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.5e+59], N[(z * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.4999999999999999e59

   1. Initial program 99.0%

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

   4. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, z, \mathsf{fma}\left(y, x, 0\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-rgt-identity 84.7

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

   6. Applied rewrites 84.7%

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

   if 2.4999999999999999e59 < z

   1. Initial program 98.0%

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

   3. Taylor expanded in x around 0

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

      1. distribute-lft1-in N/A

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

      2. metadata-eval N/A

         \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]

      4. unpow2 N/A

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

      5. lower-*.f64 98.2

         \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.2%

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

Alternative 3: 53.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot z\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (* 3.0 z) z))

C

double code(double x, double y, double z) {
	return (3.0 * 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 = (3.0d0 * z) * z
end function

Java

public static double code(double x, double y, double z) {
	return (3.0 * z) * z;
}

Python

def code(x, y, z):
	return (3.0 * z) * z

Julia

function code(x, y, z)
	return Float64(Float64(3.0 * z) * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (3.0 * z) * z;
end

Wolfram

code[x_, y_, z_] := N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]

   4. unpow2 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   5. lower-*.f64 55.6

      \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

5. Applied rewrites 55.6%

   \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right)} \]
6. Step-by-step derivation

7. Applied rewrites 55.6%

   \[\leadsto \left(3 \cdot z\right) \cdot \color{blue}{z} \]
8. Add Preprocessing

Alternative 4: 53.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(z \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 3.0 (* z z)))

C

double code(double x, double y, double z) {
	return 3.0 * (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 = 3.0d0 * (z * z)
end function

Java

public static double code(double x, double y, double z) {
	return 3.0 * (z * z);
}

Python

def code(x, y, z):
	return 3.0 * (z * z)

Julia

function code(x, y, z)
	return Float64(3.0 * Float64(z * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 3.0 * (z * z);
end

Wolfram

code[x_, y_, z_] := N[(3.0 * N[(z * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{3} \cdot {z}^{2} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]

   4. unpow2 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

   5. lower-*.f64 55.6

      \[\leadsto 3 \cdot \color{blue}{\left(z \cdot z\right)} \]

5. Applied rewrites 55.6%

   \[\leadsto \color{blue}{3 \cdot \left(z \cdot z\right)} \]
6. Add Preprocessing

Alternative 5: 32.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ z \cdot z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return 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 = z * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

4. Applied rewrites 83.5%

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

5. Taylor expanded in x around 0

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

   1. unpow2 N/A

      \[\leadsto \color{blue}{z \cdot z} \]

   2. lower-*.f64 40.5

      \[\leadsto \color{blue}{z \cdot z} \]

7. Applied rewrites 40.5%

   \[\leadsto \color{blue}{z \cdot z} \]
8. Add Preprocessing

Developer Target 1: 98.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot z\right) \cdot z + y \cdot x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (* 3.0 z) z) (* y x)))

C

double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * 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 = ((3.0d0 * z) * z) + (y * x)
end function

Java

public static double code(double x, double y, double z) {
	return ((3.0 * z) * z) + (y * x);
}

Python

def code(x, y, z):
	return ((3.0 * z) * z) + (y * x)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(3.0 * z) * z) + Float64(y * x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((3.0 * z) * z) + (y * x);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024339 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (* 3 z) z) (* y x)))

  (+ (+ (+ (* x y) (* z z)) (* z z)) (* z z)))