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

Percentage Accurate: 98.2% → 99.0%

Time: 7.1s

Alternatives: 9

Speedup: 1.8×

• 33.9% 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: 98.2% 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: 99.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. count-2 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*r* N/A

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

   8. count-2 N/A

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

   9. lower-fma.f64 N/A

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

   10. count-2 N/A

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

   11. lower-*.f64 98.4

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.1

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.1

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

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot z, z, \mathsf{fma}\left(z, z, y \cdot x\right)\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. count-2 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. distribute-lft-out N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-+.f64 99.1

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.1

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

6. Applied rewrites 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-37) (fma z (+ z z) (* y x)) (fma z (+ z z) (* z z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-37) {
		tmp = fma(z, (z + z), (y * x));
	} else {
		tmp = fma(z, (z + z), (z * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-37)
		tmp = fma(z, Float64(z + z), Float64(y * x));
	else
		tmp = fma(z, Float64(z + z), Float64(z * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-37], N[(z * N[(z + z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(z * N[(z + z), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000013e-37

   1. Initial program 99.9%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 100.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 100.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot z, z, \mathsf{fma}\left(z, z, y \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 100.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.5

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

   9. Applied rewrites 87.5%

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

   if 2.00000000000000013e-37 < (*.f64 z z)

   1. Initial program 96.2%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 96.4

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 98.1

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.1

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

   4. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot z, z, \mathsf{fma}\left(z, z, y \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 98.1

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 98.1

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

   6. Applied rewrites 98.1%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(z, z + z, \color{blue}{{z}^{2}}\right) \]
   8. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 89.6

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

   9. Applied rewrites 89.6%

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

Alternative 3: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-37) (fma z (+ z z) (* y x)) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-37) {
		tmp = fma(z, (z + z), (y * x));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-37)
		tmp = fma(z, Float64(z + z), Float64(y * x));
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-37], N[(z * N[(z + z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000013e-37

   1. Initial program 99.9%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. count-2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2 N/A

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

      11. lower-*.f64 100.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 100.0

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2 \cdot z, z, \mathsf{fma}\left(z, z, y \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. count-2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-+.f64 100.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.5

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

   9. Applied rewrites 87.5%

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

   if 2.00000000000000013e-37 < (*.f64 z z)

   1. Initial program 96.2%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 89.5

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 89.5

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

   10. Applied rewrites 89.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(z, z + z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-37) (fma z z (* y x)) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-37) {
		tmp = fma(z, z, (y * x));
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-37)
		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[N[(z * z), $MachinePrecision], 2e-37], 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 (*.f64 z z) < 2.00000000000000013e-37

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

      \[\leadsto \color{blue}{y \cdot x} + z \cdot z \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 87.2

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

   7. Applied rewrites 87.2%

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

   if 2.00000000000000013e-37 < (*.f64 z z)

   1. Initial program 96.2%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 89.5

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 89.5

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

   10. Applied rewrites 89.5%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-37}:\\ \;\;\;\;\mathsf{fma}\left(z, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-57}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-57) (* y x) (* (* 3.0 z) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-57) {
		tmp = y * x;
	} else {
		tmp = (3.0 * z) * z;
	}
	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) :: tmp
    if ((z * z) <= 2d-57) then
        tmp = y * x
    else
        tmp = (3.0d0 * z) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-57) {
		tmp = y * x;
	} else {
		tmp = (3.0 * z) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-57:
		tmp = y * x
	else:
		tmp = (3.0 * z) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-57)
		tmp = Float64(y * x);
	else
		tmp = Float64(Float64(3.0 * z) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e-57)
		tmp = y * x;
	else
		tmp = (3.0 * z) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-57], N[(y * x), $MachinePrecision], N[(N[(3.0 * z), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.99999999999999991e-57

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if 1.99999999999999991e-57 < (*.f64 z z)

   1. Initial program 96.3%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-+ N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 88.3

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

   10. Applied rewrites 88.3%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-57}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot z\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3.8 \cdot 10^{-53}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 3.8e-53) (* y x) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 3.8e-53) {
		tmp = y * x;
	} else {
		tmp = (z * z) * 3.0;
	}
	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) :: tmp
    if ((z * z) <= 3.8d-53) then
        tmp = y * x
    else
        tmp = (z * z) * 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 3.8e-53) {
		tmp = y * x;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 3.8e-53:
		tmp = y * x
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 3.8e-53)
		tmp = Float64(y * x);
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 3.8e-53)
		tmp = y * x;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 3.8e-53], N[(y * x), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.7999999999999998e-53

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if 3.7999999999999998e-53 < (*.f64 z z)

   1. Initial program 96.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

Alternative 7: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate-+l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. count-2 N/A

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

   10. distribute-lft1-in N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 99.1

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

4. Applied rewrites 99.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 99.1

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

6. Applied rewrites 99.1%

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

Alternative 8: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{3 \cdot {z}^{2} + x \cdot y} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.1

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

5. Applied rewrites 99.1%

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

6. Final simplification 99.1%

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

Alternative 9: 53.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 54.8

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

5. Applied rewrites 54.8%

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

Developer Target 1: 98.2% 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 2024270 
(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)))