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

Percentage Accurate: 98.5% → 99.6%

Time: 8.9s

Alternatives: 8

Speedup: 1.0×

• 34.0% 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: 98.5% 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: 99.6% 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 98.8%

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

   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 99.9%

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

   5. count-2 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.5% 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 99.9%

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

   4. count-2 99.9%

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

   5. distribute-rgt1-in 99.9%

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

   6. metadata-eval 99.9%

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

   7. metadata-eval 99.9%

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

   8. metadata-eval 99.9%

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

   9. *-commutative 99.9%

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

   10. associate-*l* 99.9%

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

   11. metadata-eval 99.9%

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-90} \lor \neg \left(z \cdot z \leq 5 \cdot 10^{-21}\right) \land z \cdot z \leq 5 \cdot 10^{+44}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 5e-90) (and (not (<= (* z z) 5e-21)) (<= (* z z) 5e+44)))
   (+ (* z z) (+ (* z z) (* x y)))
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 5e-90) || (!((z * z) <= 5e-21) && ((z * z) <= 5e+44))) {
		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) <= 5d-90) .or. (.not. ((z * z) <= 5d-21)) .and. ((z * z) <= 5d+44)) 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) <= 5e-90) || (!((z * z) <= 5e-21) && ((z * z) <= 5e+44))) {
		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) <= 5e-90) or (not ((z * z) <= 5e-21) and ((z * z) <= 5e+44)):
		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) <= 5e-90) || (!(Float64(z * z) <= 5e-21) && (Float64(z * z) <= 5e+44)))
		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) <= 5e-90) || (~(((z * z) <= 5e-21)) && ((z * z) <= 5e+44)))
		tmp = (z * z) + ((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], 5e-90], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 5e-21]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 5e+44]]], 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) < 5.00000000000000019e-90 or 4.99999999999999973e-21 < (*.f64 z z) < 4.9999999999999996e44

   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 89.2%

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

   if 5.00000000000000019e-90 < (*.f64 z z) < 4.99999999999999973e-21 or 4.9999999999999996e44 < (*.f64 z z)

   1. Initial program 97.6%

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

   2. Taylor expanded in x around 0 91.2%

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

   4. Simplified 91.2%

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

      1. *-un-lft-identity 91.2%

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

      2. *-un-lft-identity 91.2%

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

      3. distribute-rgt-out 91.2%

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

      4. metadata-eval 91.2%

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

      5. associate-*r* 91.2%

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

      6. *-commutative 91.2%

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

      7. fma-def 91.3%

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

      8. *-commutative 91.3%

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

      9. count-2 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. fma-udef 91.2%

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

      2. distribute-rgt-in 91.3%

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

      3. *-commutative 91.3%

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

      4. count-2 91.3%

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

      5. metadata-eval 91.3%

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

      6. distribute-lft1-in 91.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt{4} + 1\right) \cdot z\right)} \cdot z \]

      7. metadata-eval 91.3%

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

      8. metadata-eval 91.3%

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

      9. *-commutative 91.3%

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

   8. Applied egg-rr 91.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-90} \lor \neg \left(z \cdot z \leq 5 \cdot 10^{-21}\right) \land z \cdot z \leq 5 \cdot 10^{+44}:\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 4: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-90} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+25}\right) \land z \cdot z \leq 5 \cdot 10^{+44}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 5e-90) (and (not (<= (* z z) 2e+25)) (<= (* z z) 5e+44)))
   (* x y)
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 5e-90) || (!((z * z) <= 2e+25) && ((z * z) <= 5e+44))) {
		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) <= 5d-90) .or. (.not. ((z * z) <= 2d+25)) .and. ((z * z) <= 5d+44)) 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) <= 5e-90) || (!((z * z) <= 2e+25) && ((z * z) <= 5e+44))) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 5e-90) or (not ((z * z) <= 2e+25) and ((z * z) <= 5e+44)):
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(z * z) <= 5e-90) || (!(Float64(z * z) <= 2e+25) && (Float64(z * z) <= 5e+44)))
		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 * z) <= 5e-90) || (~(((z * z) <= 2e+25)) && ((z * z) <= 5e+44)))
		tmp = 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], 5e-90], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 2e+25]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 5e+44]]], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 5.00000000000000019e-90 or 2.00000000000000018e25 < (*.f64 z z) < 4.9999999999999996e44

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-def 100.0%

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

      3. associate-+l+ 100.0%

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

      4. fma-def 100.0%

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

      5. count-2 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.8%

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

      2. pow3 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*l* 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around 0 89.9%

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

   if 5.00000000000000019e-90 < (*.f64 z z) < 2.00000000000000018e25 or 4.9999999999999996e44 < (*.f64 z z)

   1. Initial program 97.7%

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

   2. Taylor expanded in x around 0 89.0%

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

   4. Simplified 89.0%

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

      1. *-un-lft-identity 89.0%

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

      2. *-un-lft-identity 89.0%

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

      3. distribute-rgt-out 89.0%

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

      4. metadata-eval 89.0%

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

      5. associate-*r* 89.0%

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

      6. *-commutative 89.0%

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

      7. fma-def 89.2%

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

      8. *-commutative 89.2%

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

      9. count-2 89.2%

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

   6. Applied egg-rr 89.2%

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

      1. fma-udef 89.0%

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

      2. distribute-rgt-in 89.1%

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

      3. *-commutative 89.1%

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

      4. count-2 89.1%

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

      5. metadata-eval 89.1%

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

      6. distribute-lft1-in 89.1%

         \[\leadsto \color{blue}{\left(\left(\sqrt{4} + 1\right) \cdot z\right)} \cdot z \]

      7. metadata-eval 89.1%

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

      8. metadata-eval 89.1%

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

      9. *-commutative 89.1%

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

   8. Applied egg-rr 89.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-90} \lor \neg \left(z \cdot z \leq 2 \cdot 10^{+25}\right) \land z \cdot z \leq 5 \cdot 10^{+44}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 5: 98.5% 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 6: 74.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.3 \cdot 10^{+189}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 1.3e+189) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1.3e+189) {
		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) <= 1.3d+189) 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) <= 1.3e+189) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1.3e+189:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1.3e+189)
		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) <= 1.3e+189)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1.3e+189], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 1.29999999999999991e189

   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. +-commutative 99.9%

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

      2. fma-def 100.0%

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

      3. associate-+l+ 100.0%

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

      4. fma-def 100.0%

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

      5. count-2 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 99.5%

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

      2. pow3 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*l* 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in z around 0 71.1%

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

   if 1.29999999999999991e189 < (*.f64 z z)

   1. Initial program 96.6%

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

   2. Taylor expanded in x around 0 98.8%

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

   4. Simplified 98.8%

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

      1. *-un-lft-identity 98.8%

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

      2. *-un-lft-identity 98.8%

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

      3. distribute-rgt-out 98.8%

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

      4. metadata-eval 98.8%

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

      5. associate-*r* 98.8%

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

      6. *-commutative 98.8%

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

      7. fma-def 98.9%

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

      8. *-commutative 98.9%

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

      9. count-2 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. fma-udef 98.8%

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

      2. distribute-rgt-in 98.8%

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

      3. *-commutative 98.8%

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

      4. count-2 98.8%

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

      5. metadata-eval 98.8%

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

      6. distribute-lft1-in 98.8%

         \[\leadsto \color{blue}{\left(\left(\sqrt{4} + 1\right) \cdot z\right)} \cdot z \]

      7. metadata-eval 98.8%

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

      8. metadata-eval 98.8%

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

      9. *-commutative 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. log1p-expm1-u 72.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(z \cdot 3\right) \cdot z\right)\right)} \]

      2. expm1-udef 72.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\left(z \cdot 3\right) \cdot z} - 1}\right) \]

      3. add-cube-cbrt 72.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{e^{\left(z \cdot 3\right) \cdot z}} \cdot \sqrt[3]{e^{\left(z \cdot 3\right) \cdot z}}\right) \cdot \sqrt[3]{e^{\left(z \cdot 3\right) \cdot z}}} - 1\right) \]

      4. fma-neg 72.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\sqrt[3]{e^{\left(z \cdot 3\right) \cdot z}} \cdot \sqrt[3]{e^{\left(z \cdot 3\right) \cdot z}}, \sqrt[3]{e^{\left(z \cdot 3\right) \cdot z}}, -1\right)}\right) \]

   10. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{z \cdot z + 0} \]
   11. Step-by-step derivation

       1. +-rgt-identity 75.0%

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

   12. Simplified 75.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 1.3 \cdot 10^{+189}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \]

Alternative 7: 53.0% 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. +-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 98.8%

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

   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 99.9%

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

   5. count-2 99.9%

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

3. Simplified 99.9%

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

   1. add-cube-cbrt 99.5%

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

   2. pow3 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in z around 0 48.2%

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

7. Final simplification 48.2%

   \[\leadsto x \cdot y \]

Alternative 8: 2.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -3.0)

C

double code(double x, double y, double z) {
	return -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 = -3.0d0
end function

Java

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

Python

def code(x, y, z):
	return -3.0

Julia

function code(x, y, z)
	return -3.0
end

MATLAB

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

Wolfram

code[x_, y_, z_] := -3.0

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 98.8%

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

   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 99.9%

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

   5. count-2 99.9%

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

3. Simplified 99.9%

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

   1. add-cube-cbrt 99.5%

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

   2. pow3 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

5. Applied egg-rr 99.5%

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

6. Taylor expanded in z around inf 35.0%

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

8. Simplified 2.1%

   \[\leadsto \color{blue}{-3} \]

9. Final simplification 2.1%

   \[\leadsto -3 \]

Developer target: 98.5% 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 2023287 
(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)))