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

Percentage Accurate: 99.9% → 99.9%

Time: 3.5s

Alternatives: 6

Speedup: 1.7×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-def 100.0%

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

   3. associate-+l+ 100.0%

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

   4. fma-def 100.0%

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

   5. distribute-lft-out 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-+l+ 99.9%

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

   3. distribute-lft-in 99.9%

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

   4. fma-udef 99.9%

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

   5. fma-udef 100.0%

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

   6. add-sqr-sqrt 99.8%

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

3. Applied egg-rr 99.8%

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

5. Simplified 99.8%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, y\right)\right)\right)\right)}^{2}} \]

6. Taylor expanded in y around 0 99.9%

   \[\leadsto \color{blue}{{x}^{2} + 3 \cdot {y}^{2}} \]
7. Step-by-step derivation

   1. metadata-eval 99.9%

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

   2. distribute-rgt1-in 99.9%

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

   3. +-commutative 99.9%

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

   4. distribute-rgt1-in 99.9%

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

   5. metadata-eval 99.9%

      \[\leadsto \color{blue}{3} \cdot {y}^{2} + {x}^{2} \]

   6. fma-def 99.9%

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

   7. unpow2 99.9%

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

   8. unpow2 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 3: 82.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4.3 \cdot 10^{+26}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 4.3e+26) (* x x) (* 3.0 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4.3e+26) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 4.3d+26) then
        tmp = x * x
    else
        tmp = 3.0d0 * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 4.3e+26) {
		tmp = x * x;
	} else {
		tmp = 3.0 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 4.3e+26:
		tmp = x * x
	else:
		tmp = 3.0 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 4.3e+26)
		tmp = Float64(x * x);
	else
		tmp = Float64(3.0 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 4.3e+26)
		tmp = x * x;
	else
		tmp = 3.0 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 4.3e+26], N[(x * x), $MachinePrecision], N[(3.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 4.2999999999999998e26

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 82.9%

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

   4. Simplified 82.9%

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

   if 4.2999999999999998e26 < (*.f64 y y)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 88.2%

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

   4. Simplified 88.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 4.3 \cdot 10^{+26}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 4: 82.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+26}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 5e+26) (* x x) (* y (* y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+26) {
		tmp = x * x;
	} else {
		tmp = y * (y * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y * y) <= 5d+26) then
        tmp = x * x
    else
        tmp = y * (y * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 5e+26) {
		tmp = x * x;
	} else {
		tmp = y * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 5e+26:
		tmp = x * x
	else:
		tmp = y * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 5e+26)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 5e+26)
		tmp = x * x;
	else
		tmp = y * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 5e+26], N[(x * x), $MachinePrecision], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 5.0000000000000001e26

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 82.9%

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

   4. Simplified 82.9%

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

   if 5.0000000000000001e26 < (*.f64 y y)

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.8%

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

      3. distribute-lft-in 99.8%

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

      4. fma-udef 99.8%

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

      5. fma-udef 99.9%

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

      6. add-sqr-sqrt 99.7%

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

   3. Applied egg-rr 99.8%

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(y, \mathsf{hypot}\left(y, \mathsf{hypot}\left(x, y\right)\right)\right)\right)}^{2}} \]

   6. Taylor expanded in x around 0 87.9%

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

      1. distribute-rgt1-in 87.9%

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

      2. unpow2 87.9%

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

      3. rem-square-sqrt 88.1%

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

      4. metadata-eval 88.1%

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

      5. unpow2 88.1%

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

   8. Simplified 88.1%

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

      1. unpow2 88.1%

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

      2. add-sqr-sqrt 88.2%

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

      3. associate-*r* 88.3%

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

      4. *-commutative 88.3%

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

   10. Applied egg-rr 88.3%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 5 \cdot 10^{+26}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 99.9%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto x \cdot x + 3 \cdot \left(y \cdot y\right) \]

Alternative 6: 57.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around inf 58.3%

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

4. Simplified 58.3%

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

5. Final simplification 58.3%

   \[\leadsto x \cdot x \]

Developer target: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023182 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

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

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