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

Percentage Accurate: 98.4% → 99.5%

Time: 7.6s

Alternatives: 7

Speedup: 1.0×

• 35.3% 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: 98.4% 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: 99.5% 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 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. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(z, z, \mathsf{fma}\left(x, y, 2 \cdot \left(z \cdot z\right)\right)\right) \]
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 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. associate-+r+ 99.9%

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

   5. distribute-lft-out 99.9%

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

   6. distribute-lft-out 99.9%

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

   7. remove-double-neg 99.9%

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

   8. unsub-neg 99.9%

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

   9. count-2 99.9%

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

   10. neg-mul-1 99.9%

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

   11. distribute-rgt-out-- 99.9%

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

   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. Add Preprocessing

5. Final simplification 99.9%

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

Alternative 3: 84.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+38} \lor \neg \left(z \cdot z \leq 10^{+109}\right) \land \left(z \cdot z \leq 5 \cdot 10^{+127} \lor \neg \left(z \cdot z \leq 5 \cdot 10^{+197}\right) \land z \cdot z \leq 4 \cdot 10^{+235}\right):\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 5e+38)
         (and (not (<= (* z z) 1e+109))
              (or (<= (* z z) 5e+127)
                  (and (not (<= (* z z) 5e+197)) (<= (* z z) 4e+235)))))
   (+ (* 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+38) || (!((z * z) <= 1e+109) && (((z * z) <= 5e+127) || (!((z * z) <= 5e+197) && ((z * z) <= 4e+235))))) {
		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+38) .or. (.not. ((z * z) <= 1d+109)) .and. ((z * z) <= 5d+127) .or. (.not. ((z * z) <= 5d+197)) .and. ((z * z) <= 4d+235)) 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+38) || (!((z * z) <= 1e+109) && (((z * z) <= 5e+127) || (!((z * z) <= 5e+197) && ((z * z) <= 4e+235))))) {
		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+38) or (not ((z * z) <= 1e+109) and (((z * z) <= 5e+127) or (not ((z * z) <= 5e+197) and ((z * z) <= 4e+235)))):
		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+38) || (!(Float64(z * z) <= 1e+109) && ((Float64(z * z) <= 5e+127) || (!(Float64(z * z) <= 5e+197) && (Float64(z * z) <= 4e+235)))))
		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+38) || (~(((z * z) <= 1e+109)) && (((z * z) <= 5e+127) || (~(((z * z) <= 5e+197)) && ((z * z) <= 4e+235)))))
		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+38], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 1e+109]], $MachinePrecision], Or[LessEqual[N[(z * z), $MachinePrecision], 5e+127], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 5e+197]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 4e+235]]]]], 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) < 4.9999999999999997e38 or 9.99999999999999982e108 < (*.f64 z z) < 5.0000000000000004e127 or 5.00000000000000009e197 < (*.f64 z z) < 4.0000000000000002e235

   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 87.8%

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

   if 4.9999999999999997e38 < (*.f64 z z) < 9.99999999999999982e108 or 5.0000000000000004e127 < (*.f64 z z) < 5.00000000000000009e197 or 4.0000000000000002e235 < (*.f64 z z)

   1. Initial program 96.8%

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

   3. Taylor expanded in x around 0 91.8%

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

   5. Simplified 91.8%

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

      1. add-sqr-sqrt 91.7%

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

      2. sqrt-unprod 69.9%

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

      3. swap-sqr 69.8%

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

      4. pow-sqr 69.9%

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

      5. metadata-eval 69.9%

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

      6. metadata-eval 69.9%

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

   7. Applied egg-rr 69.9%

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

      1. metadata-eval 69.9%

         \[\leadsto \sqrt{{z}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot 9} \]

      2. pow-sqr 69.8%

         \[\leadsto \sqrt{\color{blue}{\left({z}^{2} \cdot {z}^{2}\right)} \cdot 9} \]

      3. metadata-eval 69.8%

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

      4. swap-sqr 69.9%

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

      5. sqrt-unprod 91.7%

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

      6. add-sqr-sqrt 91.8%

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

      7. *-commutative 91.8%

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

      8. pow2 91.8%

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

      9. associate-*r* 91.8%

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

   9. Applied egg-rr 91.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+38} \lor \neg \left(z \cdot z \leq 10^{+109}\right) \land \left(z \cdot z \leq 5 \cdot 10^{+127} \lor \neg \left(z \cdot z \leq 5 \cdot 10^{+197}\right) \land z \cdot z \leq 4 \cdot 10^{+235}\right):\\ \;\;\;\;z \cdot z + \left(z \cdot z + x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+38} \lor \neg \left(z \cdot z \leq 10^{+109}\right) \land \left(z \cdot z \leq 5 \cdot 10^{+127} \lor \neg \left(z \cdot z \leq 5 \cdot 10^{+197}\right) \land z \cdot z \leq 2 \cdot 10^{+231}\right):\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 5e+38)
         (and (not (<= (* z z) 1e+109))
              (or (<= (* z z) 5e+127)
                  (and (not (<= (* z z) 5e+197)) (<= (* z z) 2e+231)))))
   (+ (* z z) (* x y))
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 5e+38) || (!((z * z) <= 1e+109) && (((z * z) <= 5e+127) || (!((z * z) <= 5e+197) && ((z * z) <= 2e+231))))) {
		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) <= 5d+38) .or. (.not. ((z * z) <= 1d+109)) .and. ((z * z) <= 5d+127) .or. (.not. ((z * z) <= 5d+197)) .and. ((z * z) <= 2d+231)) 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) <= 5e+38) || (!((z * z) <= 1e+109) && (((z * z) <= 5e+127) || (!((z * z) <= 5e+197) && ((z * z) <= 2e+231))))) {
		tmp = (z * z) + (x * y);
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 5e+38) or (not ((z * z) <= 1e+109) and (((z * z) <= 5e+127) or (not ((z * z) <= 5e+197) and ((z * z) <= 2e+231)))):
		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) <= 5e+38) || (!(Float64(z * z) <= 1e+109) && ((Float64(z * z) <= 5e+127) || (!(Float64(z * z) <= 5e+197) && (Float64(z * z) <= 2e+231)))))
		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) <= 5e+38) || (~(((z * z) <= 1e+109)) && (((z * z) <= 5e+127) || (~(((z * z) <= 5e+197)) && ((z * z) <= 2e+231)))))
		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], 5e+38], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 1e+109]], $MachinePrecision], Or[LessEqual[N[(z * z), $MachinePrecision], 5e+127], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 5e+197]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 2e+231]]]]], 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) < 4.9999999999999997e38 or 9.99999999999999982e108 < (*.f64 z z) < 5.0000000000000004e127 or 5.00000000000000009e197 < (*.f64 z z) < 2.0000000000000001e231

   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 88.7%

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

   4. Taylor expanded in x around inf 88.4%

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

   if 4.9999999999999997e38 < (*.f64 z z) < 9.99999999999999982e108 or 5.0000000000000004e127 < (*.f64 z z) < 5.00000000000000009e197 or 2.0000000000000001e231 < (*.f64 z z)

   1. Initial program 96.9%

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

   3. Taylor expanded in x around 0 90.2%

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

   5. Simplified 90.2%

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

      1. add-sqr-sqrt 90.1%

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

      2. sqrt-unprod 68.2%

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

      3. swap-sqr 68.2%

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

      4. pow-sqr 68.2%

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

      5. metadata-eval 68.2%

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

      6. metadata-eval 68.2%

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

   7. Applied egg-rr 68.2%

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

      1. metadata-eval 68.2%

         \[\leadsto \sqrt{{z}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot 9} \]

      2. pow-sqr 68.2%

         \[\leadsto \sqrt{\color{blue}{\left({z}^{2} \cdot {z}^{2}\right)} \cdot 9} \]

      3. metadata-eval 68.2%

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

      4. swap-sqr 68.2%

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

      5. sqrt-unprod 90.1%

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

      6. add-sqr-sqrt 90.2%

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

      7. *-commutative 90.2%

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

      8. pow2 90.2%

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

      9. associate-*r* 90.2%

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

   9. Applied egg-rr 90.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{+38} \lor \neg \left(z \cdot z \leq 10^{+109}\right) \land \left(z \cdot z \leq 5 \cdot 10^{+127} \lor \neg \left(z \cdot z \leq 5 \cdot 10^{+197}\right) \land z \cdot z \leq 2 \cdot 10^{+231}\right):\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1450000000 \lor \neg \left(z \leq 4.2 \cdot 10^{+54}\right) \land \left(z \leq 1.75 \cdot 10^{+65} \lor \neg \left(z \leq 3.35 \cdot 10^{+99}\right) \land z \leq 3.9 \cdot 10^{+115}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z 1450000000.0)
         (and (not (<= z 4.2e+54))
              (or (<= z 1.75e+65)
                  (and (not (<= z 3.35e+99)) (<= z 3.9e+115)))))
   (* x y)
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= 1450000000.0) || (!(z <= 4.2e+54) && ((z <= 1.75e+65) || (!(z <= 3.35e+99) && (z <= 3.9e+115))))) {
		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 <= 1450000000.0d0) .or. (.not. (z <= 4.2d+54)) .and. (z <= 1.75d+65) .or. (.not. (z <= 3.35d+99)) .and. (z <= 3.9d+115)) 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 <= 1450000000.0) || (!(z <= 4.2e+54) && ((z <= 1.75e+65) || (!(z <= 3.35e+99) && (z <= 3.9e+115))))) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= 1450000000.0) or (not (z <= 4.2e+54) and ((z <= 1.75e+65) or (not (z <= 3.35e+99) and (z <= 3.9e+115)))):
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= 1450000000.0) || (!(z <= 4.2e+54) && ((z <= 1.75e+65) || (!(z <= 3.35e+99) && (z <= 3.9e+115)))))
		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 <= 1450000000.0) || (~((z <= 4.2e+54)) && ((z <= 1.75e+65) || (~((z <= 3.35e+99)) && (z <= 3.9e+115)))))
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, 1450000000.0], And[N[Not[LessEqual[z, 4.2e+54]], $MachinePrecision], Or[LessEqual[z, 1.75e+65], And[N[Not[LessEqual[z, 3.35e+99]], $MachinePrecision], LessEqual[z, 3.9e+115]]]]], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.45e9 or 4.19999999999999972e54 < z < 1.75e65 or 3.35000000000000012e99 < z < 3.90000000000000006e115

   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. Step-by-step derivation

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

      3. fma-def 99.9%

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

      4. pow2 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in z around 0 66.2%

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

   if 1.45e9 < z < 4.19999999999999972e54 or 1.75e65 < z < 3.35000000000000012e99 or 3.90000000000000006e115 < z

   1. Initial program 95.6%

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

   3. Taylor expanded in x around 0 85.0%

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

   5. Simplified 85.0%

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

      1. add-sqr-sqrt 84.9%

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

      2. sqrt-unprod 68.8%

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

      3. swap-sqr 68.7%

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

      4. pow-sqr 68.8%

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

      5. metadata-eval 68.8%

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

      6. metadata-eval 68.8%

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

   7. Applied egg-rr 68.8%

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

      1. metadata-eval 68.8%

         \[\leadsto \sqrt{{z}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot 9} \]

      2. pow-sqr 68.7%

         \[\leadsto \sqrt{\color{blue}{\left({z}^{2} \cdot {z}^{2}\right)} \cdot 9} \]

      3. metadata-eval 68.7%

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

      4. swap-sqr 68.8%

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

      5. sqrt-unprod 84.9%

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

      6. add-sqr-sqrt 85.0%

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

      7. *-commutative 85.0%

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

      8. pow2 85.0%

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

      9. associate-*r* 84.9%

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

   9. Applied egg-rr 84.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1450000000 \lor \neg \left(z \leq 4.2 \cdot 10^{+54}\right) \land \left(z \leq 1.75 \cdot 10^{+65} \lor \neg \left(z \leq 3.35 \cdot 10^{+99}\right) \land z \leq 3.9 \cdot 10^{+115}\right):\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% 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. Add Preprocessing

3. Final simplification 98.7%

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

Alternative 7: 52.5% 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. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 98.7%

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

   2. fma-def 98.7%

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

   3. fma-def 99.9%

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

   4. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in z around 0 57.6%

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

6. Final simplification 57.6%

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

Developer target: 98.4% 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 2024021 
(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)))