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

Percentage Accurate: 98.3% → 99.3%

Time: 12.0s

Alternatives: 9

Speedup: 1.8×

• 34.9% 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: 98.3% 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 97.6%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
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 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-198}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -1e-166)
   (fma y x (* z z))
   (if (<= (* x y) 5e-198) (fma (+ z z) z (* z z)) (fma (+ z z) z (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -1e-166) {
		tmp = fma(y, x, (z * z));
	} else if ((x * y) <= 5e-198) {
		tmp = fma((z + z), z, (z * z));
	} else {
		tmp = fma((z + z), z, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= -1e-166)
		tmp = fma(y, x, Float64(z * z));
	elseif (Float64(x * y) <= 5e-198)
		tmp = fma(Float64(z + z), z, Float64(z * z));
	else
		tmp = fma(Float64(z + z), z, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e-166], N[(y * x + N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-198], N[(N[(z + z), $MachinePrecision] * z + N[(z * z), $MachinePrecision]), $MachinePrecision], N[(N[(z + z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.00000000000000004e-166

   1. Initial program 93.7%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. lower-+.f64 93.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 93.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 93.7

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

   4. Applied rewrites 93.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. flip-- N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. distribute-lft-out-- N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. +-inverses N/A

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

      21. +-rgt-identity 85.2

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

      22. lift-fma.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. +-commutative N/A

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

      25. lift-*.f64 N/A

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

      26. lower-fma.f64 91.5

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

   6. Applied rewrites 91.5%

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

   if -1.00000000000000004e-166 < (*.f64 x y) < 4.9999999999999999e-198

   1. Initial program 99.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. lower-+.f64 99.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 99.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 94.0

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

   7. Applied rewrites 94.0%

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

   if 4.9999999999999999e-198 < (*.f64 x y)

   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. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 100.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-198}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, z \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-198}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x y) -1e-166)
   (fma y x (* z z))
   (if (<= (* x y) 5e-198) (* (* z z) 3.0) (fma (+ z z) z (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * y) <= -1e-166) {
		tmp = fma(y, x, (z * z));
	} else if ((x * y) <= 5e-198) {
		tmp = (z * z) * 3.0;
	} else {
		tmp = fma((z + z), z, (x * y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * y) <= -1e-166)
		tmp = fma(y, x, Float64(z * z));
	elseif (Float64(x * y) <= 5e-198)
		tmp = Float64(Float64(z * z) * 3.0);
	else
		tmp = fma(Float64(z + z), z, Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e-166], N[(y * x + N[(z * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e-198], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision], N[(N[(z + z), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.00000000000000004e-166

   1. Initial program 93.7%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. lower-+.f64 93.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 93.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 93.7

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

   4. Applied rewrites 93.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. flip-- N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. distribute-lft-out-- N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. +-inverses N/A

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

      21. +-rgt-identity 85.2

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

      22. lift-fma.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. +-commutative N/A

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

      25. lift-*.f64 N/A

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

      26. lower-fma.f64 91.5

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

   6. Applied rewrites 91.5%

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

   if -1.00000000000000004e-166 < (*.f64 x y) < 4.9999999999999999e-198

   1. Initial program 99.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 93.9

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

   5. Applied rewrites 93.9%

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

   if 4.9999999999999999e-198 < (*.f64 x y)

   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. lower-+.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 100.0

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-166}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-198}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + z, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, x, z \cdot z\right)\\ \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-198}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y x (* z z))))
   (if (<= (* x y) -1e-166) t_0 (if (<= (* x y) 5e-198) (* (* z z) 3.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, x, (z * z));
	double tmp;
	if ((x * y) <= -1e-166) {
		tmp = t_0;
	} else if ((x * y) <= 5e-198) {
		tmp = (z * z) * 3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, x, Float64(z * z))
	tmp = 0.0
	if (Float64(x * y) <= -1e-166)
		tmp = t_0;
	elseif (Float64(x * y) <= 5e-198)
		tmp = Float64(Float64(z * z) * 3.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * x + N[(z * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -1e-166], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 5e-198], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000004e-166 or 4.9999999999999999e-198 < (*.f64 x y)

   1. Initial program 96.9%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. lower-+.f64 96.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 96.9

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 96.9

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

   4. Applied rewrites 96.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. flip-- N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. distribute-lft-out-- N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. +-inverses N/A

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

      21. +-rgt-identity 84.8

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

      22. lift-fma.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. +-commutative N/A

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

      25. lift-*.f64 N/A

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

      26. lower-fma.f64 87.9

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

   6. Applied rewrites 87.9%

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

   if -1.00000000000000004e-166 < (*.f64 x y) < 4.9999999999999999e-198

   1. Initial program 99.8%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 93.9

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

   5. Applied rewrites 93.9%

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

Alternative 5: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 3, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 5e+303) (fma (* z 3.0) z (* x y)) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 5e+303) {
		tmp = fma((z * 3.0), z, (x * y));
	} else {
		tmp = z * z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 5e+303)
		tmp = fma(Float64(z * 3.0), z, Float64(x * y));
	else
		tmp = Float64(z * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 5e+303], N[(N[(z * 3.0), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.9999999999999997e303

   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

      \[\leadsto \color{blue}{3 \cdot {z}^{2} + x \cdot y} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   if 4.9999999999999997e303 < (*.f64 z z)

   1. Initial program 89.5%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. lower-+.f64 89.5

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 89.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. flip-- N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. distribute-lft-out-- N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. +-inverses N/A

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

      21. +-rgt-identity 89.5

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

      22. lift-fma.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. +-commutative N/A

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

      25. lift-*.f64 N/A

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

      26. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 3, z, x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.4 \cdot 10^{+33}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4.4e+33) (* x y) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4.4e+33) {
		tmp = x * y;
	} 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) <= 4.4d+33) then
        tmp = x * y
    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) <= 4.4e+33) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4.4e+33:
		tmp = x * y
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4.4e+33)
		tmp = Float64(x * y);
	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) <= 4.4e+33)
		tmp = x * y;
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4.4e+33], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.39999999999999988e33

   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 87.5

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

   5. Applied rewrites 87.5%

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

   if 4.39999999999999988e33 < (*.f64 z z)

   1. Initial program 95.0%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 82.8

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

   5. Applied rewrites 82.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.4 \cdot 10^{+33}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.4 \cdot 10^{+33}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 3\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4.4e+33) (* x y) (* (* z 3.0) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4.4e+33) {
		tmp = x * y;
	} else {
		tmp = (z * 3.0) * 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) <= 4.4d+33) then
        tmp = x * y
    else
        tmp = (z * 3.0d0) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4.4e+33) {
		tmp = x * y;
	} else {
		tmp = (z * 3.0) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4.4e+33:
		tmp = x * y
	else:
		tmp = (z * 3.0) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4.4e+33)
		tmp = Float64(x * y);
	else
		tmp = Float64(Float64(z * 3.0) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 4.4e+33)
		tmp = x * y;
	else
		tmp = (z * 3.0) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4.4e+33], N[(x * y), $MachinePrecision], N[(N[(z * 3.0), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.39999999999999988e33

   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 87.5

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

   5. Applied rewrites 87.5%

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

   if 4.39999999999999988e33 < (*.f64 z z)

   1. Initial program 95.0%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{3 \cdot {z}^{2}} \]
   7. 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 82.7

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

   8. Applied rewrites 82.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.4 \cdot 10^{+33}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot 3\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 9.5 \cdot 10^{+303}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 9.5e+303) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 9.5e+303) {
		tmp = x * y;
	} else {
		tmp = 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) <= 9.5d+303) then
        tmp = x * y
    else
        tmp = z * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 9.5e+303) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 9.5e+303:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 9.5e+303)
		tmp = Float64(x * y);
	else
		tmp = Float64(z * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 9.5e+303)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 9.5e+303], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.50000000000000015e303

   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 69.4

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

   5. Applied rewrites 69.4%

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

   if 9.50000000000000015e303 < (*.f64 z z)

   1. Initial program 89.5%

      \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
   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. lower-+.f64 89.5

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 89.5

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. flip-+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. +-inverses N/A

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

      13. metadata-eval N/A

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

      14. flip-- N/A

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

      15. metadata-eval N/A

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

      16. +-inverses N/A

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

      17. distribute-lft-out-- N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. +-inverses N/A

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

      21. +-rgt-identity 89.5

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

      22. lift-fma.f64 N/A

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

      23. lift-*.f64 N/A

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

      24. +-commutative N/A

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

      25. lift-*.f64 N/A

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

      26. lower-fma.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 9.5 \cdot 10^{+303}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.5% 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 97.6%

   \[\left(\left(x \cdot y + z \cdot z\right) + z \cdot z\right) + z \cdot z \]
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. lower-+.f64 97.6

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 97.6

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 97.6

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

4. Applied rewrites 97.6%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. flip-+ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. +-inverses N/A

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

   13. metadata-eval N/A

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

   14. flip-- N/A

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

   15. metadata-eval N/A

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

   16. +-inverses N/A

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

   17. distribute-lft-out-- N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. +-inverses N/A

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

   21. +-rgt-identity 77.2

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

   22. lift-fma.f64 N/A

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

   23. lift-*.f64 N/A

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

   24. +-commutative N/A

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

   25. lift-*.f64 N/A

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

   26. lower-fma.f64 79.6

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

6. Applied rewrites 79.6%

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

7. Taylor expanded in z around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 31.3

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

9. Applied rewrites 31.3%

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

Developer Target 1: 98.3% 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 2024235 
(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)))