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

Percentage Accurate: 98.3% → 99.4%

Time: 5.3s

Alternatives: 8

Speedup: 1.1×

• 34.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. +-commutative 97.9%

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

   2. fma-def 98.0%

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

   3. associate-+l+ 98.0%

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

   4. fma-def 99.6%

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

   5. count-2 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

   1. associate-+l+ 97.9%

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

   2. associate-+l+ 97.9%

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

   3. fma-def 99.5%

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

   4. count-2 99.5%

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

   5. distribute-rgt1-in 99.5%

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

   6. metadata-eval 99.5%

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

   7. metadata-eval 99.5%

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

   8. metadata-eval 99.5%

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

   9. *-commutative 99.5%

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

   10. associate-*l* 99.5%

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

   11. metadata-eval 99.5%

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

   12. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot z + x \cdot y\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 10^{-23}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* z z) (* x y))))
   (if (<= (* z z) 2e-84)
     t_0
     (if (<= (* z z) 1e-23)
       (* (* z z) 3.0)
       (if (<= (* z z) 2e+30) t_0 (* z (* z 3.0)))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * z) + (x * y);
	double tmp;
	if ((z * z) <= 2e-84) {
		tmp = t_0;
	} else if ((z * z) <= 1e-23) {
		tmp = (z * z) * 3.0;
	} else if ((z * z) <= 2e+30) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (z * z) + (x * y)
    if ((z * z) <= 2d-84) then
        tmp = t_0
    else if ((z * z) <= 1d-23) then
        tmp = (z * z) * 3.0d0
    else if ((z * z) <= 2d+30) then
        tmp = t_0
    else
        tmp = z * (z * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * z) + (x * y);
	double tmp;
	if ((z * z) <= 2e-84) {
		tmp = t_0;
	} else if ((z * z) <= 1e-23) {
		tmp = (z * z) * 3.0;
	} else if ((z * z) <= 2e+30) {
		tmp = t_0;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * z) + (x * y)
	tmp = 0
	if (z * z) <= 2e-84:
		tmp = t_0
	elif (z * z) <= 1e-23:
		tmp = (z * z) * 3.0
	elif (z * z) <= 2e+30:
		tmp = t_0
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * z) + Float64(x * y))
	tmp = 0.0
	if (Float64(z * z) <= 2e-84)
		tmp = t_0;
	elseif (Float64(z * z) <= 1e-23)
		tmp = Float64(Float64(z * z) * 3.0);
	elseif (Float64(z * z) <= 2e+30)
		tmp = t_0;
	else
		tmp = Float64(z * Float64(z * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * z) + (x * y);
	tmp = 0.0;
	if ((z * z) <= 2e-84)
		tmp = t_0;
	elseif ((z * z) <= 1e-23)
		tmp = (z * z) * 3.0;
	elseif ((z * z) <= 2e+30)
		tmp = t_0;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * z), $MachinePrecision], 2e-84], t$95$0, If[LessEqual[N[(z * z), $MachinePrecision], 1e-23], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+30], t$95$0, N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2.0000000000000001e-84 or 9.9999999999999996e-24 < (*.f64 z z) < 2e30

   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 0 99.9%

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

   4. Simplified 100.0%

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

      1. fma-def 100.0%

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

      2. metadata-eval 100.0%

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

      3. distribute-rgt-out 100.0%

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

      4. *-commutative 100.0%

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

      5. *-un-lft-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-lft-out 99.9%

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

      8. associate-+r+ 99.9%

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

      9. +-commutative 99.9%

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

      10. fma-def 99.9%

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

      11. add-log-exp 88.7%

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

      12. exp-lft-sqr 88.7%

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

      13. log-prod 88.7%

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

      14. add-log-exp 88.9%

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

      15. add-log-exp 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around 0 91.9%

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

   if 2.0000000000000001e-84 < (*.f64 z z) < 9.9999999999999996e-24

   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 0 84.4%

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

   4. Simplified 84.4%

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

      1. distribute-lft-out 84.2%

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

      2. *-commutative 84.2%

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

      3. *-un-lft-identity 84.2%

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

      4. distribute-rgt-out 84.2%

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

      5. metadata-eval 84.2%

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

      6. associate-*r* 84.4%

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

   6. Applied egg-rr 84.4%

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

   if 2e30 < (*.f64 z z)

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0 88.3%

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

   4. Simplified 88.3%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-84}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 10^{-23}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;z \cdot z + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 4: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-84} \lor \neg \left(z \cdot z \leq 10^{-23}\right) \land z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* z z) 2e-84) (and (not (<= (* z z) 1e-23)) (<= (* z z) 2e+30)))
   (* x y)
   (* z (* z 3.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((z * z) <= 2e-84) || (!((z * z) <= 1e-23) && ((z * z) <= 2e+30))) {
		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) <= 2d-84) .or. (.not. ((z * z) <= 1d-23)) .and. ((z * z) <= 2d+30)) 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) <= 2e-84) || (!((z * z) <= 1e-23) && ((z * z) <= 2e+30))) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((z * z) <= 2e-84) or (not ((z * z) <= 1e-23) and ((z * z) <= 2e+30)):
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(z * z) <= 2e-84) || (!(Float64(z * z) <= 1e-23) && (Float64(z * z) <= 2e+30)))
		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) <= 2e-84) || (~(((z * z) <= 1e-23)) && ((z * z) <= 2e+30)))
		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], 2e-84], And[N[Not[LessEqual[N[(z * z), $MachinePrecision], 1e-23]], $MachinePrecision], LessEqual[N[(z * z), $MachinePrecision], 2e+30]]], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.0000000000000001e-84 or 9.9999999999999996e-24 < (*.f64 z z) < 2e30

   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 0 99.9%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 90.7%

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

   if 2.0000000000000001e-84 < (*.f64 z z) < 9.9999999999999996e-24 or 2e30 < (*.f64 z z)

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 88.0%

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

   4. Simplified 88.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-84} \lor \neg \left(z \cdot z \leq 10^{-23}\right) \land z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 5: 83.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-84}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 10^{-23}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e-84)
   (* x y)
   (if (<= (* z z) 1e-23)
     (* (* z z) 3.0)
     (if (<= (* z z) 2e+30) (* x y) (* z (* z 3.0))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e-84) {
		tmp = x * y;
	} else if ((z * z) <= 1e-23) {
		tmp = (z * z) * 3.0;
	} else if ((z * z) <= 2e+30) {
		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) <= 2d-84) then
        tmp = x * y
    else if ((z * z) <= 1d-23) then
        tmp = (z * z) * 3.0d0
    else if ((z * z) <= 2d+30) 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) <= 2e-84) {
		tmp = x * y;
	} else if ((z * z) <= 1e-23) {
		tmp = (z * z) * 3.0;
	} else if ((z * z) <= 2e+30) {
		tmp = x * y;
	} else {
		tmp = z * (z * 3.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e-84:
		tmp = x * y
	elif (z * z) <= 1e-23:
		tmp = (z * z) * 3.0
	elif (z * z) <= 2e+30:
		tmp = x * y
	else:
		tmp = z * (z * 3.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e-84)
		tmp = Float64(x * y);
	elseif (Float64(z * z) <= 1e-23)
		tmp = Float64(Float64(z * z) * 3.0);
	elseif (Float64(z * z) <= 2e+30)
		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) <= 2e-84)
		tmp = x * y;
	elseif ((z * z) <= 1e-23)
		tmp = (z * z) * 3.0;
	elseif ((z * z) <= 2e+30)
		tmp = x * y;
	else
		tmp = z * (z * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e-84], N[(x * y), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 1e-23], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[N[(z * z), $MachinePrecision], 2e+30], N[(x * y), $MachinePrecision], N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z z) < 2.0000000000000001e-84 or 9.9999999999999996e-24 < (*.f64 z z) < 2e30

   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 0 99.9%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in z around 0 90.7%

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

   if 2.0000000000000001e-84 < (*.f64 z z) < 9.9999999999999996e-24

   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 0 84.4%

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

   4. Simplified 84.4%

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

      1. distribute-lft-out 84.2%

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

      2. *-commutative 84.2%

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

      3. *-un-lft-identity 84.2%

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

      4. distribute-rgt-out 84.2%

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

      5. metadata-eval 84.2%

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

      6. associate-*r* 84.4%

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

   6. Applied egg-rr 84.4%

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

   if 2e30 < (*.f64 z z)

   1. Initial program 95.6%

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

   2. Taylor expanded in x around 0 88.3%

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

   4. Simplified 88.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-84}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;z \cdot z \leq 10^{-23}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \mathbf{elif}\;z \cdot z \leq 2 \cdot 10^{+30}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right)\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+245}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right) + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 2e+245) (+ (* z (* z 3.0)) (* x y)) (* (* z z) 3.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 2e+245) {
		tmp = (z * (z * 3.0)) + (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+245) then
        tmp = (z * (z * 3.0d0)) + (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+245) {
		tmp = (z * (z * 3.0)) + (x * y);
	} else {
		tmp = (z * z) * 3.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 2e+245:
		tmp = (z * (z * 3.0)) + (x * y)
	else:
		tmp = (z * z) * 3.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 2e+245)
		tmp = Float64(Float64(z * Float64(z * 3.0)) + Float64(x * y));
	else
		tmp = Float64(Float64(z * z) * 3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 2e+245)
		tmp = (z * (z * 3.0)) + (x * y);
	else
		tmp = (z * z) * 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 2e+245], N[(N[(z * N[(z * 3.0), $MachinePrecision]), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(z * z), $MachinePrecision] * 3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 2.00000000000000009e245

   1. Initial program 99.8%

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

      1. associate-+l+ 99.8%

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

      2. associate-+l+ 99.8%

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

      3. fma-def 99.8%

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

      4. count-2 99.8%

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

      5. distribute-rgt1-in 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-commutative 99.8%

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

      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. Step-by-step derivation

      1. fma-udef 99.9%

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

      2. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 2.00000000000000009e245 < (*.f64 z z)

   1. Initial program 92.7%

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

   2. Taylor expanded in x around 0 98.5%

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

   4. Simplified 98.5%

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

      1. distribute-lft-out 98.5%

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

      2. *-commutative 98.5%

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

      3. *-un-lft-identity 98.5%

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

      4. distribute-rgt-out 98.5%

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

      5. metadata-eval 98.5%

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

      6. associate-*r* 98.5%

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

   6. Applied egg-rr 98.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+245}:\\ \;\;\;\;z \cdot \left(z \cdot 3\right) + x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot z\right) \cdot 3\\ \end{array} \]

Alternative 7: 74.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.2 \cdot 10^{+248}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (if (<= (* z z) 4.2e+248) (* x y) (* z z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 4.2e+248) {
		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) <= 4.2d+248) 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) <= 4.2e+248) {
		tmp = x * y;
	} else {
		tmp = z * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4.2e+248:
		tmp = x * y
	else:
		tmp = z * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 4.2e+248)
		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) <= 4.2e+248)
		tmp = x * y;
	else
		tmp = z * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 4.2e+248], N[(x * y), $MachinePrecision], N[(z * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 4.19999999999999977e248

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in z around 0 70.6%

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

   if 4.19999999999999977e248 < (*.f64 z z)

   1. Initial program 92.7%

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

   2. Taylor expanded in x around 0 92.7%

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

   4. Simplified 94.1%

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

      1. fma-def 92.7%

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

      2. metadata-eval 92.7%

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

      3. distribute-rgt-out 92.7%

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

      4. *-commutative 92.7%

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

      5. *-un-lft-identity 92.7%

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

      6. +-commutative 92.7%

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

      7. distribute-lft-out 92.7%

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

      8. associate-+r+ 92.7%

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

      9. +-commutative 92.7%

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

      10. fma-def 92.7%

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

      11. add-log-exp 81.1%

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

      12. exp-lft-sqr 81.1%

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

      13. log-prod 81.1%

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

      14. add-log-exp 81.1%

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

      15. add-log-exp 92.7%

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

   6. Applied egg-rr 92.7%

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

   7. Taylor expanded in z around 0 82.0%

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

   8. Taylor expanded in x around 0 87.8%

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

   10. Simplified 87.8%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4.2 \cdot 10^{+248}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot z\\ \end{array} \]

Alternative 8: 52.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.9%

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

2. Taylor expanded in x around 0 97.9%

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

4. Simplified 98.3%

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

5. Taylor expanded in z around 0 52.8%

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

6. Final simplification 52.8%

   \[\leadsto x \cdot y \]

Developer target: 98.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023293 
(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)))