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

Percentage Accurate: 98.3% → 98.3%

Time: 7.6s

Alternatives: 7

Speedup: 0.1×

• 35.0% 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: 98.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. fma-define N/A

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

   5. fma-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(z, \color{blue}{\left(z + z\right)}, \left(x \cdot y + z \cdot z\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{z}\right), \left(x \cdot y + z \cdot z\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{+.f64}\left(z, z\right), \mathsf{+.f64}\left(\left(x \cdot y\right), \left(z \cdot z\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{+.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z\right)\right)\right) \]

   9. *-lowering-*.f64 97.6%

      \[\leadsto \mathsf{fma.f64}\left(z, \mathsf{+.f64}\left(z, z\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, z\right)\right)\right) \]

4. Applied egg-rr 97.6%

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

Alternative 2: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-10}:\\ \;\;\;\;x \cdot y + z \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{0.3333333333333333}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-10) (+ (* x y) (* z z)) (/ z (/ 0.3333333333333333 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-10) {
		tmp = (x * y) + (z * z);
	} else {
		tmp = z / (0.3333333333333333 / 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) <= 1d-10) then
        tmp = (x * y) + (z * z)
    else
        tmp = z / (0.3333333333333333d0 / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-10) {
		tmp = (x * y) + (z * z);
	} else {
		tmp = z / (0.3333333333333333 / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-10:
		tmp = (x * y) + (z * z)
	else:
		tmp = z / (0.3333333333333333 / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-10)
		tmp = Float64(Float64(x * y) + Float64(z * z));
	else
		tmp = Float64(z / Float64(0.3333333333333333 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e-10)
		tmp = (x * y) + (z * z);
	else
		tmp = z / (0.3333333333333333 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-10], N[(N[(x * y), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision], N[(z / N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e-10

   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 x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, \mathsf{*.f64}\left(z, z\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 90.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{z}, z\right)\right) \]

   5. Simplified 90.0%

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

   if 1.00000000000000004e-10 < (*.f64 z z)

   1. Initial program 95.3%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 95.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{x \cdot y + \left(z \cdot z\right) \cdot 3} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot 3\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot 3\right) \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot 3\right), \color{blue}{z}\right)\right) \]

      4. *-lowering-*.f64 95.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, 3\right), z\right)\right) \]

   6. Applied egg-rr 95.3%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 3\right) \cdot z\right) \cdot \left(\left(z \cdot 3\right) \cdot z\right) - \left(x \cdot y\right) \cdot \left(\left(z \cdot 3\right) \cdot z\right)\right)}{{\left(x \cdot y\right)}^{3} + {\left(\left(z \cdot 3\right) \cdot z\right)}^{3}}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{\left(x \cdot y\right)}^{3} + {\left(\left(z \cdot 3\right) \cdot z\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 3\right) \cdot z\right) \cdot \left(\left(z \cdot 3\right) \cdot z\right) - \left(x \cdot y\right) \cdot \left(\left(z \cdot 3\right) \cdot z\right)\right)}}}\right)\right) \]

      5. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{x \cdot y + \color{blue}{\left(z \cdot 3\right) \cdot z}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot y + \left(z \cdot 3\right) \cdot z\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot 3\right) \cdot z\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot 3\right)} \cdot z\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot 3\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot 3\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 95.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{3}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 95.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{3}}{{z}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

       3. *-lowering-*.f64 85.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   11. Simplified 85.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{0.3333333333333333}{z \cdot z}}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{\frac{\frac{1}{3}}{z}}{\color{blue}{z}}} \]

       2. clear-num N/A

          \[\leadsto \frac{z}{\color{blue}{\frac{\frac{1}{3}}{z}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{\frac{1}{3}}{z}\right)}\right) \]

       4. /-lowering-/.f64 85.6%

          \[\leadsto \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{z}\right)\right) \]

   13. Applied egg-rr 85.6%

       \[\leadsto \color{blue}{\frac{z}{\frac{0.3333333333333333}{z}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{-10}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{0.3333333333333333}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-10) (* x y) (/ z (/ 0.3333333333333333 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-10) {
		tmp = x * y;
	} else {
		tmp = z / (0.3333333333333333 / 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) <= 1d-10) then
        tmp = x * y
    else
        tmp = z / (0.3333333333333333d0 / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-10) {
		tmp = x * y;
	} else {
		tmp = z / (0.3333333333333333 / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-10:
		tmp = x * y
	else:
		tmp = z / (0.3333333333333333 / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-10)
		tmp = Float64(x * y);
	else
		tmp = Float64(z / Float64(0.3333333333333333 / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e-10)
		tmp = x * y;
	else
		tmp = z / (0.3333333333333333 / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e-10], N[(x * y), $MachinePrecision], N[(z / N[(0.3333333333333333 / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e-10

   1. Initial program 99.9%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 88.6%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 88.6%

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

   if 1.00000000000000004e-10 < (*.f64 z z)

   1. Initial program 95.3%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 95.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 95.3%

      \[\leadsto \color{blue}{x \cdot y + \left(z \cdot z\right) \cdot 3} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot 3\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot 3\right) \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot 3\right), \color{blue}{z}\right)\right) \]

      4. *-lowering-*.f64 95.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, 3\right), z\right)\right) \]

   6. Applied egg-rr 95.3%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 3\right) \cdot z\right) \cdot \left(\left(z \cdot 3\right) \cdot z\right) - \left(x \cdot y\right) \cdot \left(\left(z \cdot 3\right) \cdot z\right)\right)}{{\left(x \cdot y\right)}^{3} + {\left(\left(z \cdot 3\right) \cdot z\right)}^{3}}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{\left(x \cdot y\right)}^{3} + {\left(\left(z \cdot 3\right) \cdot z\right)}^{3}}{\left(x \cdot y\right) \cdot \left(x \cdot y\right) + \left(\left(\left(z \cdot 3\right) \cdot z\right) \cdot \left(\left(z \cdot 3\right) \cdot z\right) - \left(x \cdot y\right) \cdot \left(\left(z \cdot 3\right) \cdot z\right)\right)}}}\right)\right) \]

      5. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{x \cdot y + \color{blue}{\left(z \cdot 3\right) \cdot z}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(x \cdot y + \left(z \cdot 3\right) \cdot z\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(\left(z \cdot 3\right) \cdot z\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{\left(z \cdot 3\right)} \cdot z\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot 3\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot 3\right)}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 95.2%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, \color{blue}{3}\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 95.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{3}}{{z}^{2}}\right)}\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left({z}^{2}\right)}\right)\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \left(z \cdot \color{blue}{z}\right)\right)\right) \]

       3. *-lowering-*.f64 85.5%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right) \]

   11. Simplified 85.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{0.3333333333333333}{z \cdot z}}} \]
   12. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \frac{1}{\frac{\frac{\frac{1}{3}}{z}}{\color{blue}{z}}} \]

       2. clear-num N/A

          \[\leadsto \frac{z}{\color{blue}{\frac{\frac{1}{3}}{z}}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(z, \color{blue}{\left(\frac{\frac{1}{3}}{z}\right)}\right) \]

       4. /-lowering-/.f64 85.6%

          \[\leadsto \mathsf{/.f64}\left(z, \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{z}\right)\right) \]

   13. Applied egg-rr 85.6%

       \[\leadsto \color{blue}{\frac{z}{\frac{0.3333333333333333}{z}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e-10) (* x y) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e-10) {
		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) <= 1d-10) 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) <= 1e-10) {
		tmp = x * y;
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e-10:
		tmp = x * y
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e-10)
		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) <= 1e-10)
		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], 1e-10], N[(x * y), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.00000000000000004e-10

   1. Initial program 99.9%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 88.6%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 88.6%

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

   if 1.00000000000000004e-10 < (*.f64 z z)

   1. Initial program 95.3%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 95.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 95.3%

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

   5. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \color{blue}{\left({z}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(3, \left(z \cdot \color{blue}{z}\right)\right) \]

      3. *-lowering-*.f64 85.6%

         \[\leadsto \mathsf{*.f64}\left(3, \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right) \]

   7. Simplified 85.6%

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

4. Final simplification 87.0%

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

Alternative 5: 74.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 3.15 \cdot 10^{+262}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 3.15e+262) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 3.15e+262) {
		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) <= 3.15d+262) 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) <= 3.15e+262) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 3.15e+262:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 3.15e+262)
		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) <= 3.15e+262)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 3.15e+262], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.15000000000000002e262

   1. Initial program 99.8%

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

      1. associate-+l+ N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

      5. count-2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      6. distribute-rgt1-in N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

      10. metadata-eval 99.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 70.3%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 70.3%

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

   if 3.15000000000000002e262 < (*.f64 z z)

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(x \cdot y\right)}, \mathsf{*.f64}\left(z, z\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 82.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\color{blue}{z}, z\right)\right) \]

   5. Simplified 82.4%

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

   6. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 86.3%

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{z}\right) \]

   8. Simplified 86.3%

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

Alternative 6: 98.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) + (z * (z * 3.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

   10. metadata-eval 97.5%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

3. Simplified 97.5%

   \[\leadsto \color{blue}{x \cdot y + \left(z \cdot z\right) \cdot 3} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot \color{blue}{\left(z \cdot 3\right)}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot 3\right) \cdot \color{blue}{z}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot 3\right), \color{blue}{z}\right)\right) \]

   4. *-lowering-*.f64 97.5%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, 3\right), z\right)\right) \]

6. Applied egg-rr 97.5%

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

7. Final simplification 97.5%

   \[\leadsto x \cdot y + z \cdot \left(z \cdot 3\right) \]
8. Add Preprocessing

Alternative 7: 52.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

   1. associate-+l+ N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot z + \left(z \cdot z + z \cdot z\right)\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z \cdot z} + \left(z \cdot z + z \cdot z\right)\right)\right) \]

   5. count-2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(z \cdot z + 2 \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   6. distribute-rgt1-in N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(2 + 1\right) \cdot \color{blue}{\left(z \cdot z\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\left(z \cdot z\right) \cdot \color{blue}{\left(2 + 1\right)}\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\left(z \cdot z\right), \color{blue}{\left(2 + 1\right)}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(\color{blue}{2} + 1\right)\right)\right) \]

   10. metadata-eval 97.5%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, z\right), 3\right)\right) \]

3. Simplified 97.5%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.f64 52.3%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

7. Simplified 52.3%

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

Developer Target 1: 98.3% accurate, 1.7× 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 2024158 
(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)))