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

Percentage Accurate: 98.2% → 99.4%

Time: 13.5s

Alternatives: 8

Speedup: 0.7×

• 34.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z z (fma x y (* 2.0 (* z z)))))

C

double code(double x, double y, double z) {
	return fma(z, z, fma(x, y, (2.0 * (z * z))));
}

Julia

function code(x, y, z)
	return fma(z, z, fma(x, y, Float64(2.0 * Float64(z * z))))
end

Wolfram

code[x_, y_, z_] := N[(z * z + N[(x * y + N[(2.0 * N[(z * z), $MachinePrecision]), $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. +-commutative 97.5%

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

   2. fma-define 97.6%

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

   3. associate-+l+ 97.6%

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

   4. fma-define 99.2%

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

   5. count-2 99.2%

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

3. Simplified 99.2%

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

5. Final simplification 99.2%

   \[\leadsto \mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right) \]
6. Add Preprocessing

Alternative 2: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(x * y + 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+ 97.6%

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

   2. associate-+l+ 97.5%

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

   3. fma-define 99.1%

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

   4. associate-+r+ 99.1%

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

   5. distribute-lft-out 99.1%

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

   6. distribute-lft-out 99.1%

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

   7. remove-double-neg 99.1%

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

   8. unsub-neg 99.1%

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

   9. count-2 99.1%

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

   10. neg-mul-1 99.1%

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

   11. distribute-rgt-out-- 99.1%

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

   12. metadata-eval 99.1%

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

3. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+294}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+294)
   (+ (* z z) (+ (* z z) (+ (* z z) (* x y))))
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+294) {
		tmp = (z * z) + ((z * z) + ((z * z) + (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) <= 2d+294) then
        tmp = (z * z) + ((z * z) + ((z * z) + (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) <= 2e+294) {
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+294:
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)))
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+294)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y))));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+294)
		tmp = (z * z) + ((z * z) + ((z * z) + (x * y)));
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+294], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000013e294

   1. Initial program 99.9%

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

   if 2.00000000000000013e294 < (*.f64 z z)

   1. Initial program 91.3%

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

   3. Taylor expanded in x around 0 97.1%

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

      1. distribute-lft1-in 97.1%

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

      2. metadata-eval 97.1%

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

      3. *-commutative 97.1%

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

   5. Simplified 97.1%

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

      1. add-sqr-sqrt 97.1%

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

      2. sqrt-unprod 94.6%

         \[\leadsto \color{blue}{\sqrt{\left({z}^{2} \cdot 3\right) \cdot \left({z}^{2} \cdot 3\right)}} \]

      3. swap-sqr 94.6%

         \[\leadsto \sqrt{\color{blue}{\left({z}^{2} \cdot {z}^{2}\right) \cdot \left(3 \cdot 3\right)}} \]

      4. pow-prod-up 94.6%

         \[\leadsto \sqrt{\color{blue}{{z}^{\left(2 + 2\right)}} \cdot \left(3 \cdot 3\right)} \]

      5. metadata-eval 94.6%

         \[\leadsto \sqrt{{z}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)} \]

      6. metadata-eval 94.6%

         \[\leadsto \sqrt{{z}^{4} \cdot \color{blue}{9}} \]

   7. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. *-commutative 94.6%

         \[\leadsto \sqrt{\color{blue}{9 \cdot {z}^{4}}} \]

      2. sqrt-prod 94.6%

         \[\leadsto \color{blue}{\sqrt{9} \cdot \sqrt{{z}^{4}}} \]

      3. metadata-eval 94.6%

         \[\leadsto \color{blue}{3} \cdot \sqrt{{z}^{4}} \]

      4. metadata-eval 94.6%

         \[\leadsto \color{blue}{{3}^{1}} \cdot \sqrt{{z}^{4}} \]

      5. metadata-eval 94.6%

         \[\leadsto {3}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{{z}^{4}} \]

      6. sqrt-pow2 94.6%

         \[\leadsto \color{blue}{{\left(\sqrt{3}\right)}^{2}} \cdot \sqrt{{z}^{4}} \]

      7. sqrt-pow1 97.0%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}} \]

      8. metadata-eval 97.0%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot {z}^{\color{blue}{2}} \]

      9. unpow2 97.0%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

      10. associate-*r* 97.1%

          \[\leadsto \color{blue}{\left({\left(\sqrt{3}\right)}^{2} \cdot z\right) \cdot z} \]

      11. sqrt-pow2 97.1%

          \[\leadsto \left(\color{blue}{{3}^{\left(\frac{2}{2}\right)}} \cdot z\right) \cdot z \]

      12. metadata-eval 97.1%

          \[\leadsto \left({3}^{\color{blue}{1}} \cdot z\right) \cdot z \]

      13. metadata-eval 97.1%

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

   9. Applied egg-rr 97.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+294}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + \left(z \cdot z + x \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-57} \lor \neg \left(z \leq 2.1 \cdot 10^{-41}\right) \land z \leq 0.155:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 6.2e-57) (and (not (<= z 2.1e-41)) (<= z 0.155)))
   (* x y)
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 6.2e-57) || (!(z <= 2.1e-41) && (z <= 0.155))) {
		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 <= 6.2d-57) .or. (.not. (z <= 2.1d-41)) .and. (z <= 0.155d0)) 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 <= 6.2e-57) || (!(z <= 2.1e-41) && (z <= 0.155))) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 6.2e-57) or (not (z <= 2.1e-41) and (z <= 0.155)):
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 6.2e-57) || (!(z <= 2.1e-41) && (z <= 0.155)))
		tmp = Float64(x * y);
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= 6.2e-57) || (~((z <= 2.1e-41)) && (z <= 0.155)))
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 6.2e-57], And[N[Not[LessEqual[z, 2.1e-41]], $MachinePrecision], LessEqual[z, 0.155]]], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.19999999999999952e-57 or 2.10000000000000013e-41 < z < 0.154999999999999999

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf 87.2%

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

   4. Taylor expanded in x around inf 86.8%

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

   5. Taylor expanded in x around inf 71.6%

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

   if 6.19999999999999952e-57 < z < 2.10000000000000013e-41 or 0.154999999999999999 < z

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0 91.7%

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

      1. distribute-lft1-in 91.7%

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

      2. metadata-eval 91.7%

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

      3. *-commutative 91.7%

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

   5. Simplified 91.7%

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

      1. add-sqr-sqrt 91.5%

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

      2. sqrt-unprod 78.8%

         \[\leadsto \color{blue}{\sqrt{\left({z}^{2} \cdot 3\right) \cdot \left({z}^{2} \cdot 3\right)}} \]

      3. swap-sqr 78.8%

         \[\leadsto \sqrt{\color{blue}{\left({z}^{2} \cdot {z}^{2}\right) \cdot \left(3 \cdot 3\right)}} \]

      4. pow-prod-up 78.8%

         \[\leadsto \sqrt{\color{blue}{{z}^{\left(2 + 2\right)}} \cdot \left(3 \cdot 3\right)} \]

      5. metadata-eval 78.8%

         \[\leadsto \sqrt{{z}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)} \]

      6. metadata-eval 78.8%

         \[\leadsto \sqrt{{z}^{4} \cdot \color{blue}{9}} \]

   7. Applied egg-rr 78.8%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. *-commutative 78.8%

         \[\leadsto \sqrt{\color{blue}{9 \cdot {z}^{4}}} \]

      2. sqrt-prod 78.8%

         \[\leadsto \color{blue}{\sqrt{9} \cdot \sqrt{{z}^{4}}} \]

      3. metadata-eval 78.8%

         \[\leadsto \color{blue}{3} \cdot \sqrt{{z}^{4}} \]

      4. metadata-eval 78.8%

         \[\leadsto \color{blue}{{3}^{1}} \cdot \sqrt{{z}^{4}} \]

      5. metadata-eval 78.8%

         \[\leadsto {3}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{{z}^{4}} \]

      6. sqrt-pow2 78.6%

         \[\leadsto \color{blue}{{\left(\sqrt{3}\right)}^{2}} \cdot \sqrt{{z}^{4}} \]

      7. sqrt-pow1 91.3%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}} \]

      8. metadata-eval 91.3%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot {z}^{\color{blue}{2}} \]

      9. unpow2 91.3%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

      10. associate-*r* 91.3%

          \[\leadsto \color{blue}{\left({\left(\sqrt{3}\right)}^{2} \cdot z\right) \cdot z} \]

      11. sqrt-pow2 91.7%

          \[\leadsto \left(\color{blue}{{3}^{\left(\frac{2}{2}\right)}} \cdot z\right) \cdot z \]

      12. metadata-eval 91.7%

          \[\leadsto \left({3}^{\color{blue}{1}} \cdot z\right) \cdot z \]

      13. metadata-eval 91.7%

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

   9. Applied egg-rr 91.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.2 \cdot 10^{-57} \lor \neg \left(z \leq 2.1 \cdot 10^{-41}\right) \land z \leq 0.155:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.001) (+ (* z z) (+ (* z z) (* x y))) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.001) {
		tmp = (z * z) + ((z * z) + (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) <= 0.001d0) then
        tmp = (z * z) + ((z * z) + (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) <= 0.001) {
		tmp = (z * z) + ((z * z) + (x * y));
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.001:
		tmp = (z * z) + ((z * z) + (x * y))
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.001)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y)));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.001)
		tmp = (z * z) + ((z * z) + (x * y));
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.001], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-3

   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 91.8%

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

   if 1e-3 < (*.f64 z z)

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0 88.1%

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

      1. distribute-lft1-in 88.1%

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

      2. metadata-eval 88.1%

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

      3. *-commutative 88.1%

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

   5. Simplified 88.1%

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

      1. add-sqr-sqrt 87.9%

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

      2. sqrt-unprod 75.9%

         \[\leadsto \color{blue}{\sqrt{\left({z}^{2} \cdot 3\right) \cdot \left({z}^{2} \cdot 3\right)}} \]

      3. swap-sqr 75.9%

         \[\leadsto \sqrt{\color{blue}{\left({z}^{2} \cdot {z}^{2}\right) \cdot \left(3 \cdot 3\right)}} \]

      4. pow-prod-up 75.9%

         \[\leadsto \sqrt{\color{blue}{{z}^{\left(2 + 2\right)}} \cdot \left(3 \cdot 3\right)} \]

      5. metadata-eval 75.9%

         \[\leadsto \sqrt{{z}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)} \]

      6. metadata-eval 75.9%

         \[\leadsto \sqrt{{z}^{4} \cdot \color{blue}{9}} \]

   7. Applied egg-rr 75.9%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. *-commutative 75.9%

         \[\leadsto \sqrt{\color{blue}{9 \cdot {z}^{4}}} \]

      2. sqrt-prod 75.9%

         \[\leadsto \color{blue}{\sqrt{9} \cdot \sqrt{{z}^{4}}} \]

      3. metadata-eval 75.9%

         \[\leadsto \color{blue}{3} \cdot \sqrt{{z}^{4}} \]

      4. metadata-eval 75.9%

         \[\leadsto \color{blue}{{3}^{1}} \cdot \sqrt{{z}^{4}} \]

      5. metadata-eval 75.9%

         \[\leadsto {3}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{{z}^{4}} \]

      6. sqrt-pow2 75.7%

         \[\leadsto \color{blue}{{\left(\sqrt{3}\right)}^{2}} \cdot \sqrt{{z}^{4}} \]

      7. sqrt-pow1 87.7%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}} \]

      8. metadata-eval 87.7%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot {z}^{\color{blue}{2}} \]

      9. unpow2 87.7%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

      10. associate-*r* 87.8%

          \[\leadsto \color{blue}{\left({\left(\sqrt{3}\right)}^{2} \cdot z\right) \cdot z} \]

      11. sqrt-pow2 88.1%

          \[\leadsto \left(\color{blue}{{3}^{\left(\frac{2}{2}\right)}} \cdot z\right) \cdot z \]

      12. metadata-eval 88.1%

          \[\leadsto \left({3}^{\color{blue}{1}} \cdot z\right) \cdot z \]

      13. metadata-eval 88.1%

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

   9. Applied egg-rr 88.1%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.001)
   (+ (* z z) (+ (* z z) (* x y)))
   (+ (* z z) (+ (* z z) (* z z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.001) {
		tmp = (z * z) + ((z * z) + (x * y));
	} else {
		tmp = (z * z) + ((z * z) + (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) <= 0.001d0) then
        tmp = (z * z) + ((z * z) + (x * y))
    else
        tmp = (z * z) + ((z * z) + (z * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.001) {
		tmp = (z * z) + ((z * z) + (x * y));
	} else {
		tmp = (z * z) + ((z * z) + (z * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.001:
		tmp = (z * z) + ((z * z) + (x * y))
	else:
		tmp = (z * z) + ((z * z) + (z * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.001)
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(x * y)));
	else
		tmp = Float64(Float64(z * z) + Float64(Float64(z * z) + Float64(z * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.001)
		tmp = (z * z) + ((z * z) + (x * y));
	else
		tmp = (z * z) + ((z * z) + (z * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.001], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] + N[(N[(z * z), $MachinePrecision] + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-3

   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 91.8%

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

   if 1e-3 < (*.f64 z z)

   1. Initial program 94.9%

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

      1. add-cube-cbrt 94.6%

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

      2. pow3 94.6%

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

      3. fma-define 97.9%

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

      4. pow2 97.9%

         \[\leadsto \left({\left(\sqrt[3]{\mathsf{fma}\left(x, y, \color{blue}{{z}^{2}}\right)}\right)}^{3} + z \cdot z\right) + z \cdot z \]

   4. Applied egg-rr 97.9%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, y, {z}^{2}\right)}\right)}^{3}} + z \cdot z\right) + z \cdot z \]
   5. Step-by-step derivation

      1. rem-cbrt-cube 97.8%

         \[\leadsto \left({\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(x, y, {z}^{2}\right)}\right)}^{3}}\right)}}^{3} + z \cdot z\right) + z \cdot z \]

      2. add-cube-cbrt 97.4%

         \[\leadsto \left({\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(x, y, {z}^{2}\right)}\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(x, y, {z}^{2}\right)}\right)}^{3}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(x, y, {z}^{2}\right)}\right)}^{3}}}\right)}}^{3} + z \cdot z\right) + z \cdot z \]

      3. pow3 97.4%

         \[\leadsto \left({\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(x, y, {z}^{2}\right)}\right)}^{3}}}\right)}^{3}\right)}}^{3} + z \cdot z\right) + z \cdot z \]

      4. rem-cbrt-cube 97.5%

         \[\leadsto \left({\left({\left(\sqrt[3]{\color{blue}{\sqrt[3]{\mathsf{fma}\left(x, y, {z}^{2}\right)}}}\right)}^{3}\right)}^{3} + z \cdot z\right) + z \cdot z \]

   6. Applied egg-rr 97.5%

      \[\leadsto \left({\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(x, y, {z}^{2}\right)}}\right)}^{3}\right)}}^{3} + z \cdot z\right) + z \cdot z \]

   7. Taylor expanded in x around 0 85.7%

      \[\leadsto \left({\left({\left(\sqrt[3]{\color{blue}{{\left({z}^{2}\right)}^{0.3333333333333333}}}\right)}^{3}\right)}^{3} + z \cdot z\right) + z \cdot z \]
   8. Step-by-step derivation

      1. unpow1/3 87.6%

         \[\leadsto \left({\left({\left(\sqrt[3]{\color{blue}{\sqrt[3]{{z}^{2}}}}\right)}^{3}\right)}^{3} + z \cdot z\right) + z \cdot z \]

   9. Simplified 87.6%

      \[\leadsto \left({\left({\left(\sqrt[3]{\color{blue}{\sqrt[3]{{z}^{2}}}}\right)}^{3}\right)}^{3} + z \cdot z\right) + z \cdot z \]
   10. Step-by-step derivation

       1. unpow3 87.5%

          \[\leadsto \left({\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{{z}^{2}}} \cdot \sqrt[3]{\sqrt[3]{{z}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{{z}^{2}}}\right)}}^{3} + z \cdot z\right) + z \cdot z \]

       2. add-cube-cbrt 87.9%

          \[\leadsto \left({\color{blue}{\left(\sqrt[3]{{z}^{2}}\right)}}^{3} + z \cdot z\right) + z \cdot z \]

       3. rem-cube-cbrt 88.1%

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

       4. unpow2 88.1%

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

   11. Applied egg-rr 88.1%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + z \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 0.001) (+ (* z z) (* x y)) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 0.001) {
		tmp = (z * z) + (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) <= 0.001d0) then
        tmp = (z * z) + (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) <= 0.001) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 0.001:
		tmp = (z * z) + (x * y)
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 0.001)
		tmp = Float64(Float64(z * z) + Float64(x * y));
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 0.001)
		tmp = (z * z) + (x * y);
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 0.001], N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1e-3

   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 91.8%

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

   4. Taylor expanded in x around inf 91.6%

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

   if 1e-3 < (*.f64 z z)

   1. Initial program 94.9%

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

   3. Taylor expanded in x around 0 88.1%

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

      1. distribute-lft1-in 88.1%

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

      2. metadata-eval 88.1%

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

      3. *-commutative 88.1%

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

   5. Simplified 88.1%

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

      1. add-sqr-sqrt 87.9%

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

      2. sqrt-unprod 75.9%

         \[\leadsto \color{blue}{\sqrt{\left({z}^{2} \cdot 3\right) \cdot \left({z}^{2} \cdot 3\right)}} \]

      3. swap-sqr 75.9%

         \[\leadsto \sqrt{\color{blue}{\left({z}^{2} \cdot {z}^{2}\right) \cdot \left(3 \cdot 3\right)}} \]

      4. pow-prod-up 75.9%

         \[\leadsto \sqrt{\color{blue}{{z}^{\left(2 + 2\right)}} \cdot \left(3 \cdot 3\right)} \]

      5. metadata-eval 75.9%

         \[\leadsto \sqrt{{z}^{\color{blue}{4}} \cdot \left(3 \cdot 3\right)} \]

      6. metadata-eval 75.9%

         \[\leadsto \sqrt{{z}^{4} \cdot \color{blue}{9}} \]

   7. Applied egg-rr 75.9%

      \[\leadsto \color{blue}{\sqrt{{z}^{4} \cdot 9}} \]
   8. Step-by-step derivation

      1. *-commutative 75.9%

         \[\leadsto \sqrt{\color{blue}{9 \cdot {z}^{4}}} \]

      2. sqrt-prod 75.9%

         \[\leadsto \color{blue}{\sqrt{9} \cdot \sqrt{{z}^{4}}} \]

      3. metadata-eval 75.9%

         \[\leadsto \color{blue}{3} \cdot \sqrt{{z}^{4}} \]

      4. metadata-eval 75.9%

         \[\leadsto \color{blue}{{3}^{1}} \cdot \sqrt{{z}^{4}} \]

      5. metadata-eval 75.9%

         \[\leadsto {3}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \sqrt{{z}^{4}} \]

      6. sqrt-pow2 75.7%

         \[\leadsto \color{blue}{{\left(\sqrt{3}\right)}^{2}} \cdot \sqrt{{z}^{4}} \]

      7. sqrt-pow1 87.7%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot \color{blue}{{z}^{\left(\frac{4}{2}\right)}} \]

      8. metadata-eval 87.7%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot {z}^{\color{blue}{2}} \]

      9. unpow2 87.7%

         \[\leadsto {\left(\sqrt{3}\right)}^{2} \cdot \color{blue}{\left(z \cdot z\right)} \]

      10. associate-*r* 87.8%

          \[\leadsto \color{blue}{\left({\left(\sqrt{3}\right)}^{2} \cdot z\right) \cdot z} \]

      11. sqrt-pow2 88.1%

          \[\leadsto \left(\color{blue}{{3}^{\left(\frac{2}{2}\right)}} \cdot z\right) \cdot z \]

      12. metadata-eval 88.1%

          \[\leadsto \left({3}^{\color{blue}{1}} \cdot z\right) \cdot z \]

      13. metadata-eval 88.1%

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

   9. Applied egg-rr 88.1%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 0.001:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.1% 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. Add Preprocessing

3. Taylor expanded in x around inf 80.0%

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

4. Taylor expanded in x around inf 79.5%

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

5. Taylor expanded in x around inf 55.8%

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

6. Final simplification 55.8%

   \[\leadsto x \cdot y \]
7. Add Preprocessing

Developer target: 98.2% 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 2024039 
(FPCore (x y z)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (+ (* (* 3.0 z) z) (* y x))

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