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

Percentage Accurate: 99.9% → 100.0%

Time: 7.0s

Alternatives: 8

Speedup: 1.0×

• 44.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 100.0% 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 99.8%

   \[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
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. 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+ 99.9%

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

   4. fma-def 99.9%

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

   5. distribute-lft-out 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

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 99.8%

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

   1. associate-+l+ 99.9%

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

   2. associate-+l+ 99.9%

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

   3. fma-def 99.9%

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

   4. cancel-sign-sub 99.9%

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

   5. count-2 99.9%

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

   6. neg-mul-1 99.9%

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

   7. associate-*l* 99.9%

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

   8. distribute-rgt-out-- 99.9%

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

   9. 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: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-136} \lor \neg \left(x \cdot x \leq 10^{-50}\right) \land x \cdot x \leq 100000000:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 1e-136)
         (and (not (<= (* x x) 1e-50)) (<= (* x x) 100000000.0)))
   (* y (* y 3.0))
   (* x x)))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 1e-136) || (!((x * x) <= 1e-50) && ((x * x) <= 100000000.0))) {
		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) <= 1d-136) .or. (.not. ((x * x) <= 1d-50)) .and. ((x * x) <= 100000000.0d0)) 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) <= 1e-136) || (!((x * x) <= 1e-50) && ((x * x) <= 100000000.0))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 1e-136) or (not ((x * x) <= 1e-50) and ((x * x) <= 100000000.0)):
		tmp = y * (y * 3.0)
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 1e-136) || (!(Float64(x * x) <= 1e-50) && (Float64(x * x) <= 100000000.0)))
		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) <= 1e-136) || (~(((x * x) <= 1e-50)) && ((x * x) <= 100000000.0)))
		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], 1e-136], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 1e-50]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 100000000.0]]], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(x * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e-136 or 1.00000000000000001e-50 < (*.f64 x x) < 1e8

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 88.3%

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

      1. unpow2 88.3%

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

      2. unpow2 88.3%

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

      3. distribute-lft1-in 88.3%

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

      4. metadata-eval 88.3%

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

      5. *-commutative 88.3%

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

      6. associate-*l* 88.4%

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

   4. Simplified 88.4%

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

   if 1e-136 < (*.f64 x x) < 1.00000000000000001e-50 or 1e8 < (*.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 85.8%

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

      1. unpow2 85.8%

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

   4. Simplified 85.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-136} \lor \neg \left(x \cdot x \leq 10^{-50}\right) \land x \cdot x \leq 100000000:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

Alternative 4: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-136}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{-50}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 100000000:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e-136)
   (* y (* y 3.0))
   (if (<= (* x x) 1e-50)
     (* x x)
     (if (<= (* x x) 100000000.0) (* (* y y) 3.0) (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-136) {
		tmp = y * (y * 3.0);
	} else if ((x * x) <= 1e-50) {
		tmp = x * x;
	} else if ((x * x) <= 100000000.0) {
		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) <= 1d-136) then
        tmp = y * (y * 3.0d0)
    else if ((x * x) <= 1d-50) then
        tmp = x * x
    else if ((x * x) <= 100000000.0d0) 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) <= 1e-136) {
		tmp = y * (y * 3.0);
	} else if ((x * x) <= 1e-50) {
		tmp = x * x;
	} else if ((x * x) <= 100000000.0) {
		tmp = (y * y) * 3.0;
	} else {
		tmp = x * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 1e-136:
		tmp = y * (y * 3.0)
	elif (x * x) <= 1e-50:
		tmp = x * x
	elif (x * x) <= 100000000.0:
		tmp = (y * y) * 3.0
	else:
		tmp = x * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-136)
		tmp = Float64(y * Float64(y * 3.0));
	elseif (Float64(x * x) <= 1e-50)
		tmp = Float64(x * x);
	elseif (Float64(x * x) <= 100000000.0)
		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) <= 1e-136)
		tmp = y * (y * 3.0);
	elseif ((x * x) <= 1e-50)
		tmp = x * x;
	elseif ((x * x) <= 100000000.0)
		tmp = (y * y) * 3.0;
	else
		tmp = x * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-136], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e-50], N[(x * x), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 100000000.0], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision], N[(x * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1e-136

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 89.4%

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

      1. unpow2 89.4%

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

      2. unpow2 89.4%

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

      3. distribute-lft1-in 89.4%

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

      4. metadata-eval 89.4%

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

      5. *-commutative 89.4%

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

      6. associate-*l* 89.5%

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

   4. Simplified 89.5%

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

   if 1e-136 < (*.f64 x x) < 1.00000000000000001e-50 or 1e8 < (*.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 85.8%

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

      1. unpow2 85.8%

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

   4. Simplified 85.8%

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

   if 1.00000000000000001e-50 < (*.f64 x x) < 1e8

   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. add-sqr-sqrt_binary64 99.7%

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

   3. Applied rewrite-once 99.7%

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

   4. Taylor expanded in x around 0 80.5%

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

      1. distribute-lft1-in 80.5%

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

      2. metadata-eval 80.5%

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

      3. *-commutative 80.5%

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

      4. unpow2 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-136}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \cdot x \leq 10^{-50}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;x \cdot x \leq 100000000:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{else}:\\ \;\;\;\;x \cdot x\\ \end{array} \]

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 99.8%

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

2. Final simplification 99.8%

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

Alternative 6: 9.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-144}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.35e-144) 0.0 x))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.35e-144) {
		tmp = 0.0;
	} else {
		tmp = 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 <= 1.35d-144) then
        tmp = 0.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.35e-144) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.35e-144:
		tmp = 0.0
	else:
		tmp = x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.35e-144)
		tmp = 0.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.35e-144)
		tmp = 0.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.35e-144], 0.0, x]

Derivation

1. Split input into 2 regimes

2. if x < 1.34999999999999988e-144

   1. Initial program 99.8%

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

   3. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{x - x} \]
   4. Step-by-step derivation

      1. +-inverses 8.0%

         \[\leadsto \color{blue}{0} \]

   5. Simplified 8.0%

      \[\leadsto \color{blue}{0} \]

   if 1.34999999999999988e-144 < x

   1. Initial program 99.9%

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

   3. Applied egg-rr 6.4%

      \[\leadsto \color{blue}{e^{\log x}} \]
   4. Step-by-step derivation

      1. rem-exp-log 6.4%

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

   5. Simplified 6.4%

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

4. Final simplification 7.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-144}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 58.0% 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 99.8%

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

2. Taylor expanded in x around inf 56.3%

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

   1. unpow2 56.3%

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

4. Simplified 56.3%

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

5. Final simplification 56.3%

   \[\leadsto x \cdot x \]

Alternative 8: 8.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 99.8%

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

3. Applied egg-rr 6.2%

   \[\leadsto \color{blue}{x - x} \]
4. Step-by-step derivation

   1. +-inverses 6.2%

      \[\leadsto \color{blue}{0} \]

5. Simplified 6.2%

   \[\leadsto \color{blue}{0} \]

6. Final simplification 6.2%

   \[\leadsto 0 \]

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 2023297 
(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)))