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

Percentage Accurate: 98.1% → 99.3%

Time: 6.5s

Alternatives: 8

Speedup: 1.0×

• 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: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 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 98.7%

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

   1. +-commutative 98.7%

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

   2. fma-def 98.7%

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

   3. associate-+l+ 98.8%

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

   4. fma-def 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. Final simplification 99.2%

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

Alternative 2: 99.3% 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 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. fma-def 99.1%

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

   4. count-2 99.1%

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

   5. distribute-rgt1-in 99.1%

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

   6. metadata-eval 99.1%

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

   7. metadata-eval 99.1%

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

   8. metadata-eval 99.1%

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

   9. *-commutative 99.1%

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

   10. associate-*l* 99.1%

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

   11. metadata-eval 99.1%

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

   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. Final simplification 99.1%

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

Alternative 3: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-47} \lor \neg \left(z \cdot z \leq 10^{-24}\right) \land z \cdot z \leq 1:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 2e-47) (and (not (<= (* z z) 1e-24)) (<= (* z z) 1.0)))
   (+ (* z z) (* x y))
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 2e-47) || (!((z * z) <= 1e-24) && ((z * z) <= 1.0))) {
		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) <= 2d-47) .or. (.not. ((z * z) <= 1d-24)) .and. ((z * z) <= 1.0d0)) 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) <= 2e-47) || (!((z * z) <= 1e-24) && ((z * z) <= 1.0))) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 2e-47) or (not ((z * z) <= 1e-24) and ((z * z) <= 1.0)):
		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) <= 2e-47) || (!(Float64(z * z) <= 1e-24) && (Float64(z * z) <= 1.0)))
		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) <= 2e-47) || (~(((z * z) <= 1e-24)) && ((z * z) <= 1.0)))
		tmp = (z * z) + (x * y);
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(z * z), $MachinePrecision], 2e-47], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 1e-24]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 1.0]]], 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) < 1.9999999999999999e-47 or 9.99999999999999924e-25 < (*.f64 z z) < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 92.9%

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

   3. Taylor expanded in x around inf 92.7%

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

   if 1.9999999999999999e-47 < (*.f64 z z) < 9.99999999999999924e-25 or 1 < (*.f64 z z)

   1. Initial program 97.4%

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

      1. associate-+l+ 97.5%

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

      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-def 98.2%

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

      4. count-2 98.2%

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

      5. distribute-rgt1-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. metadata-eval 98.2%

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

      8. metadata-eval 98.2%

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

      9. *-commutative 98.2%

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

      10. associate-*l* 98.2%

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

      11. metadata-eval 98.2%

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

      12. metadata-eval 98.2%

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

   3. Simplified 98.2%

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

      1. add-sqr-sqrt 98.1%

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

      2. pow2 98.1%

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

      3. associate-*r* 98.0%

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

      4. sqrt-prod 98.0%

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

      5. sqrt-prod 45.4%

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

      6. add-sqr-sqrt 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in x around 0 83.3%

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

      1. unpow2 83.3%

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

      2. unpow2 83.3%

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

      3. swap-sqr 83.4%

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

      4. unpow2 83.4%

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

   8. Simplified 83.4%

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

      1. add-sqr-sqrt 36.6%

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

      2. sqrt-prod 83.4%

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

      3. unpow2 83.4%

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

      4. sqrt-prod 83.5%

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

      5. pow2 83.5%

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

      6. add-sqr-sqrt 83.7%

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

      7. *-commutative 83.7%

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

      8. unpow2 83.7%

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

      9. associate-*r* 83.6%

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

   10. Applied egg-rr 83.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-47} \lor \neg \left(z \cdot z \leq 10^{-24}\right) \land z \cdot z \leq 1:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 4: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1.0) (+ (* z z) (+ (* z z) (* x y))) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1.0) {
		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) <= 1.0d0) 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) <= 1.0) {
		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) <= 1.0:
		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) <= 1.0)
		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) <= 1.0)
		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], 1.0], 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) < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 90.1%

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

   if 1 < (*.f64 z z)

   1. Initial program 97.3%

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

      1. associate-+l+ 97.4%

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

      2. associate-+l+ 97.4%

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

      3. fma-def 98.2%

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

      4. count-2 98.2%

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

      5. distribute-rgt1-in 98.2%

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

      6. metadata-eval 98.2%

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

      7. metadata-eval 98.2%

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

      8. metadata-eval 98.2%

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

      9. *-commutative 98.2%

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

      10. associate-*l* 98.1%

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

      11. metadata-eval 98.1%

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

      12. metadata-eval 98.1%

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

   3. Simplified 98.1%

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

      1. add-sqr-sqrt 98.0%

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

      2. pow2 98.0%

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

      3. associate-*r* 98.0%

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

      4. sqrt-prod 97.9%

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

      5. sqrt-prod 45.6%

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

      6. add-sqr-sqrt 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 82.7%

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

      1. unpow2 82.7%

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

      2. unpow2 82.7%

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

      3. swap-sqr 82.8%

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

      4. unpow2 82.8%

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

   8. Simplified 82.8%

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

      1. add-sqr-sqrt 36.5%

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

      2. sqrt-prod 82.8%

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

      3. unpow2 82.8%

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

      4. sqrt-prod 82.8%

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

      5. pow2 82.8%

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

      6. add-sqr-sqrt 83.0%

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

      7. *-commutative 83.0%

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

      8. unpow2 83.0%

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

      9. associate-*r* 83.0%

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

   10. Applied egg-rr 83.0%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 5: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + z \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1.0)
   (+ (* 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) <= 1.0) {
		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) <= 1.0d0) 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) <= 1.0) {
		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) <= 1.0:
		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) <= 1.0)
		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) <= 1.0)
		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], 1.0], 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) < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 90.1%

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

   if 1 < (*.f64 z z)

   1. Initial program 97.3%

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

      1. add-cube-cbrt 96.9%

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

      2. pow3 97.0%

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

      3. fma-def 97.8%

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

      4. pow2 97.8%

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

   3. Applied egg-rr 97.8%

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

   4. Taylor expanded in x around 0 80.1%

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

      1. unpow1/3 82.8%

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

   6. Simplified 82.8%

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

      1. pow2 82.8%

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

      2. rem-cube-cbrt 83.0%

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

   8. Applied egg-rr 83.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + z \cdot z\right)\\ \end{array} \]

Alternative 6: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

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

2. Final simplification 98.7%

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

Alternative 7: 67.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-21}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.15e-21) (* x y) (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.15e-21) {
		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 <= 1.15d-21) 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 <= 1.15e-21) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.15e-21:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.15e-21)
		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 <= 1.15e-21)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.15e-21], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.15e-21

   1. Initial program 98.9%

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

      1. associate-+l+ 98.9%

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

      2. associate-+l+ 98.9%

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

      3. fma-def 99.4%

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

      4. count-2 99.4%

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

      5. distribute-rgt1-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-*l* 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. associate-*r* 99.3%

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

      4. sqrt-prod 99.2%

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

      5. sqrt-prod 31.7%

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

      6. add-sqr-sqrt 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around inf 65.9%

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

   if 1.15e-21 < z

   1. Initial program 98.2%

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

      1. associate-+l+ 98.3%

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

      2. associate-+l+ 98.3%

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

      3. fma-def 98.3%

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

      4. count-2 98.3%

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

      5. distribute-rgt1-in 98.3%

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

      6. metadata-eval 98.3%

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

      7. metadata-eval 98.3%

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

      8. metadata-eval 98.3%

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

      9. *-commutative 98.3%

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

      10. associate-*l* 98.3%

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

      11. metadata-eval 98.3%

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

      12. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

      1. add-sqr-sqrt 98.1%

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

      2. pow2 98.1%

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

      3. associate-*r* 98.1%

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

      4. sqrt-prod 98.0%

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

      5. sqrt-prod 97.8%

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

      6. add-sqr-sqrt 98.0%

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

   5. Applied egg-rr 98.0%

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

   6. Taylor expanded in x around 0 74.6%

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

      1. unpow2 74.6%

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

      2. unpow2 74.6%

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

      3. swap-sqr 74.6%

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

      4. unpow2 74.6%

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

   8. Simplified 74.6%

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

      1. add-sqr-sqrt 74.4%

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

      2. sqrt-prod 74.6%

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

      3. unpow2 74.6%

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

      4. sqrt-prod 74.7%

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

      5. pow2 74.7%

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

      6. add-sqr-sqrt 74.9%

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

      7. *-commutative 74.9%

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

      8. unpow2 74.9%

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

      9. associate-*r* 74.8%

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

   10. Applied egg-rr 74.8%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-21}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 8: 51.9% 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 98.7%

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

   1. associate-+l+ 98.7%

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

   2. associate-+l+ 98.7%

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

   3. fma-def 99.1%

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

   4. count-2 99.1%

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

   5. distribute-rgt1-in 99.1%

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

   6. metadata-eval 99.1%

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

   7. metadata-eval 99.1%

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

   8. metadata-eval 99.1%

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

   9. *-commutative 99.1%

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

   10. associate-*l* 99.1%

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

   11. metadata-eval 99.1%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 99.0%

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

   2. pow2 99.0%

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

   3. associate-*r* 99.0%

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

   4. sqrt-prod 98.9%

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

   5. sqrt-prod 48.2%

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

   6. add-sqr-sqrt 98.9%

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

5. Applied egg-rr 98.9%

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

6. Taylor expanded in x around inf 55.5%

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

7. Final simplification 55.5%

   \[\leadsto x \cdot y \]

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 2023333 
(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)))