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

Percentage Accurate: 99.9% → 99.9%

Time: 5.2s

Alternatives: 10

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 99.9%

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

   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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(x * x + N[(N[(y * y), $MachinePrecision] * 3.0), $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. associate-+l+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. fma-def 99.9%

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

   4. count-2 99.9%

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

   5. distribute-rgt1-in 99.9%

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

   6. *-commutative 99.9%

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

   7. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.8 \cdot 10^{-169} \lor \neg \left(x \cdot x \leq 2.35 \cdot 10^{-24}\right) \land x \cdot x \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 7.8e-169)
         (and (not (<= (* x x) 2.35e-24)) (<= (* x x) 2.7e+23)))
   (* y (* y 3.0))
   (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 7.8e-169) || (!((x * x) <= 2.35e-24) && ((x * x) <= 2.7e+23))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * x) <= 7.8d-169) .or. (.not. ((x * x) <= 2.35d-24)) .and. ((x * x) <= 2.7d+23)) then
        tmp = y * (y * 3.0d0)
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * x) <= 7.8e-169) || (!((x * x) <= 2.35e-24) && ((x * x) <= 2.7e+23))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 7.8e-169) or (not ((x * x) <= 2.35e-24) and ((x * x) <= 2.7e+23)):
		tmp = y * (y * 3.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 7.8e-169) || (!(Float64(x * x) <= 2.35e-24) && (Float64(x * x) <= 2.7e+23)))
		tmp = Float64(y * Float64(y * 3.0));
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * x) <= 7.8e-169) || (~(((x * x) <= 2.35e-24)) && ((x * x) <= 2.7e+23)))
		tmp = y * (y * 3.0);
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 7.8e-169], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 2.35e-24]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 2.7e+23]]], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 7.79999999999999954e-169 or 2.34999999999999996e-24 < (*.f64 x x) < 2.6999999999999999e23

   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 87.7%

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

      1. unpow2 87.7%

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

      2. unpow2 87.7%

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

      3. distribute-rgt1-in 87.7%

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

      4. metadata-eval 87.7%

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

      5. *-commutative 87.7%

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

      6. associate-*r* 87.6%

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

   4. Simplified 87.6%

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

   if 7.79999999999999954e-169 < (*.f64 x x) < 2.34999999999999996e-24 or 2.6999999999999999e23 < (*.f64 x x)

   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 83.3%

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

      1. unpow2 83.3%

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

   4. Simplified 83.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 7.8 \cdot 10^{-169} \lor \neg \left(x \cdot x \leq 2.35 \cdot 10^{-24}\right) \land x \cdot x \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.45 \cdot 10^{-169} \lor \neg \left(x \cdot x \leq 2.35 \cdot 10^{-24}\right) \land x \cdot x \leq 5.7 \cdot 10^{+23}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 1.45e-169)
         (and (not (<= (* x x) 2.35e-24)) (<= (* x x) 5.7e+23)))
   (* (* y y) 3.0)
   (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 1.45e-169) || (!((x * x) <= 2.35e-24) && ((x * x) <= 5.7e+23))) {
		tmp = (y * y) * 3.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * x) <= 1.45d-169) .or. (.not. ((x * x) <= 2.35d-24)) .and. ((x * x) <= 5.7d+23)) then
        tmp = (y * y) * 3.0d0
    else
        tmp = x * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * x) <= 1.45e-169) || (!((x * x) <= 2.35e-24) && ((x * x) <= 5.7e+23))) {
		tmp = (y * y) * 3.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 1.45e-169) or (not ((x * x) <= 2.35e-24) and ((x * x) <= 5.7e+23)):
		tmp = (y * y) * 3.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 1.45e-169) || (!(Float64(x * x) <= 2.35e-24) && (Float64(x * x) <= 5.7e+23)))
		tmp = Float64(Float64(y * y) * 3.0);
	else
		tmp = Float64(x * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * x) <= 1.45e-169) || (~(((x * x) <= 2.35e-24)) && ((x * x) <= 5.7e+23)))
		tmp = (y * y) * 3.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 1.45e-169], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 2.35e-24]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 5.7e+23]]], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.4500000000000001e-169 or 2.34999999999999996e-24 < (*.f64 x x) < 5.7e23

   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 87.7%

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

      1. unpow2 87.7%

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

   4. Simplified 87.7%

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

      1. count-2 87.7%

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

      2. metadata-eval 87.7%

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

      3. *-un-lft-identity 87.7%

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

      4. distribute-rgt-out 87.7%

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

      5. metadata-eval 87.7%

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

      6. metadata-eval 87.7%

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

   6. Applied egg-rr 87.7%

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

   if 1.4500000000000001e-169 < (*.f64 x x) < 2.34999999999999996e-24 or 5.7e23 < (*.f64 x x)

   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 83.3%

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

      1. unpow2 83.3%

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

   4. Simplified 83.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.45 \cdot 10^{-169} \lor \neg \left(x \cdot x \leq 2.35 \cdot 10^{-24}\right) \land x \cdot x \leq 5.7 \cdot 10^{+23}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * y), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(N[(y * y), $MachinePrecision] + N[(x * x), $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. Final simplification 99.9%

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

Alternative 6: 90.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.2 \cdot 10^{-171}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4.2e-171) (* (* y y) 3.0) (+ (* y y) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4.2e-171) {
		tmp = (y * y) * 3.0;
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 4.2d-171) then
        tmp = (y * y) * 3.0d0
    else
        tmp = (y * y) + (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4.2e-171) {
		tmp = (y * y) * 3.0;
	} else {
		tmp = (y * y) + (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 4.2e-171:
		tmp = (y * y) * 3.0
	else:
		tmp = (y * y) + (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 4.2e-171)
		tmp = Float64(Float64(y * y) * 3.0);
	else
		tmp = Float64(Float64(y * y) + Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 4.2e-171)
		tmp = (y * y) * 3.0;
	else
		tmp = (y * y) + (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 4.2e-171], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision], N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.2e-171

   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%

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

      1. unpow2 88.2%

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

   4. Simplified 88.2%

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

      1. count-2 88.2%

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

      2. metadata-eval 88.2%

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

      3. *-un-lft-identity 88.2%

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

      4. distribute-rgt-out 88.2%

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

      5. metadata-eval 88.2%

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

      6. metadata-eval 88.2%

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

   6. Applied egg-rr 88.2%

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

   if 4.2e-171 < (*.f64 x x)

   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. associate-+l+ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. fma-def 100.0%

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

      4. count-2 100.0%

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

      5. distribute-rgt1-in 100.0%

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

      6. *-commutative 100.0%

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

      7. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*l* 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{y \cdot \left(y \cdot 3\right) + x \cdot x} \]
   6. Step-by-step derivation

      1. associate-*r* 32.4%

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

      2. metadata-eval 32.4%

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

      3. metadata-eval 32.4%

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

      4. distribute-rgt-out 32.4%

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

      5. metadata-eval 32.4%

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

      6. count-2 32.4%

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

      7. *-un-lft-identity 32.4%

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

      8. flip-+ 0.0%

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

      9. distribute-lft-out-- 0.0%

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

      10. +-inverses 0.0%

          \[\leadsto y \cdot y + \frac{\left(y \cdot y\right) \cdot \color{blue}{0}}{y \cdot y - y \cdot y} \]

      11. metadata-eval 0.0%

          \[\leadsto y \cdot y + \frac{\left(y \cdot y\right) \cdot \color{blue}{\left(1 - 1\right)}}{y \cdot y - y \cdot y} \]

      12. distribute-rgt-out-- 0.0%

          \[\leadsto y \cdot y + \frac{\color{blue}{1 \cdot \left(y \cdot y\right) - 1 \cdot \left(y \cdot y\right)}}{y \cdot y - y \cdot y} \]

      13. *-un-lft-identity 0.0%

          \[\leadsto y \cdot y + \frac{\color{blue}{y \cdot y} - 1 \cdot \left(y \cdot y\right)}{y \cdot y - y \cdot y} \]

      14. *-un-lft-identity 0.0%

          \[\leadsto y \cdot y + \frac{y \cdot y - \color{blue}{y \cdot y}}{y \cdot y - y \cdot y} \]

      15. +-inverses 0.0%

          \[\leadsto y \cdot y + \frac{\color{blue}{0}}{y \cdot y - y \cdot y} \]

      16. +-inverses 0.0%

          \[\leadsto y \cdot y + \frac{0}{\color{blue}{0}} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\left(y \cdot y + \frac{0}{0}\right)} + x \cdot x \]

   9. Simplified 90.9%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4.2 \cdot 10^{-171}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;y \cdot y + x \cdot x\\ \end{array} \]

Alternative 7: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 3.0), $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. associate-+l+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. fma-def 99.9%

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

   4. count-2 99.9%

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

   5. distribute-rgt1-in 99.9%

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

   6. *-commutative 99.9%

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

   7. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. fma-udef 99.9%

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

   2. +-commutative 99.9%

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

   3. associate-*l* 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 8: 75.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 9.5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 9.5e+305) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 9.5e+305) {
		tmp = x * x;
	} else {
		tmp = 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) <= 9.5d+305) then
        tmp = x * x
    else
        tmp = y * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y * y) <= 9.5e+305) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 9.5e+305:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 9.5e+305)
		tmp = Float64(x * x);
	else
		tmp = Float64(y * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * y) <= 9.5e+305)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 9.5e+305], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 9.49999999999999962e305

   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 68.8%

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

      1. unpow2 68.8%

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

   4. Simplified 68.8%

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

   if 9.49999999999999962e305 < (*.f64 y y)

   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 0 100.0%

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

      1. unpow2 100.0%

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

      2. unpow2 100.0%

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

      3. distribute-rgt1-in 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

      1. associate-*r* 100.0%

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

      2. metadata-eval 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. metadata-eval 100.0%

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

      6. count-2 100.0%

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

      7. *-un-lft-identity 100.0%

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

      8. flip-+ 0.0%

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

      9. distribute-lft-out-- 0.0%

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

      10. +-inverses 0.0%

          \[\leadsto y \cdot y + \frac{\left(y \cdot y\right) \cdot \color{blue}{0}}{y \cdot y - y \cdot y} \]

      11. metadata-eval 0.0%

          \[\leadsto y \cdot y + \frac{\left(y \cdot y\right) \cdot \color{blue}{\left(1 - 1\right)}}{y \cdot y - y \cdot y} \]

      12. distribute-rgt-out-- 0.0%

          \[\leadsto y \cdot y + \frac{\color{blue}{1 \cdot \left(y \cdot y\right) - 1 \cdot \left(y \cdot y\right)}}{y \cdot y - y \cdot y} \]

      13. *-un-lft-identity 0.0%

          \[\leadsto y \cdot y + \frac{\color{blue}{y \cdot y} - 1 \cdot \left(y \cdot y\right)}{y \cdot y - y \cdot y} \]

      14. *-un-lft-identity 0.0%

          \[\leadsto y \cdot y + \frac{y \cdot y - \color{blue}{y \cdot y}}{y \cdot y - y \cdot y} \]

      15. +-inverses 0.0%

          \[\leadsto y \cdot y + \frac{\color{blue}{0}}{y \cdot y - y \cdot y} \]

      16. +-inverses 0.0%

          \[\leadsto y \cdot y + \frac{0}{\color{blue}{0}} \]

   6. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{y \cdot y + \frac{0}{0}} \]

   8. Simplified 100.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 9.5 \cdot 10^{+305}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 9: 57.1% 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 61.1%

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

   1. unpow2 61.1%

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

4. Simplified 61.1%

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

5. Final simplification 61.1%

   \[\leadsto x \cdot x \]

Alternative 10: 1.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -2.0)

C

double code(double x, double y) {
	return -2.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -2.0d0
end function

Java

public static double code(double x, double y) {
	return -2.0;
}

Python

def code(x, y):
	return -2.0

Julia

function code(x, y)
	return -2.0
end

MATLAB

function tmp = code(x, y)
	tmp = -2.0;
end

Wolfram

code[x_, y_] := -2.0

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 52.4%

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

   1. unpow2 52.4%

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

   2. unpow2 52.4%

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

   3. distribute-rgt1-in 52.4%

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

   4. metadata-eval 52.4%

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

   5. *-commutative 52.4%

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

   6. associate-*r* 52.4%

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

4. Simplified 52.4%

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

5. Taylor expanded in y around 0 52.4%

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

7. Simplified 1.3%

   \[\leadsto \color{blue}{-2} \]

8. Final simplification 1.3%

   \[\leadsto -2 \]

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 2023252 
(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)))