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

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

Alternatives: 8

Speedup: 1.7×

• 43.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] + N[(N[(y * 2.0), $MachinePrecision] * y + N[(y * y), $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. Taylor expanded in x around 0 99.8%

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

4. Simplified 99.9%

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

   1. associate-*r* 99.8%

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

   2. metadata-eval 99.8%

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

   3. distribute-rgt-out 99.8%

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

   4. associate-*r* 99.8%

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

   5. *-un-lft-identity 99.8%

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

   6. fma-def 99.9%

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

   7. *-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 89.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.55 \cdot 10^{-193} \lor \neg \left(x \cdot x \leq 3.2 \cdot 10^{-155}\right) \land x \cdot x \leq 9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* x x) 1.55e-193)
         (and (not (<= (* x x) 3.2e-155)) (<= (* x x) 9e-57)))
   (* y (* y 3.0))
   (+ (* x x) (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (((x * x) <= 1.55e-193) || (!((x * x) <= 3.2e-155) && ((x * x) <= 9e-57))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = (x * x) + (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 (((x * x) <= 1.55d-193) .or. (.not. ((x * x) <= 3.2d-155)) .and. ((x * x) <= 9d-57)) then
        tmp = y * (y * 3.0d0)
    else
        tmp = (x * x) + (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x * x) <= 1.55e-193) || (!((x * x) <= 3.2e-155) && ((x * x) <= 9e-57))) {
		tmp = y * (y * 3.0);
	} else {
		tmp = (x * x) + (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x * x) <= 1.55e-193) or (not ((x * x) <= 3.2e-155) and ((x * x) <= 9e-57)):
		tmp = y * (y * 3.0)
	else:
		tmp = (x * x) + (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(x * x) <= 1.55e-193) || (!(Float64(x * x) <= 3.2e-155) && (Float64(x * x) <= 9e-57)))
		tmp = Float64(y * Float64(y * 3.0));
	else
		tmp = Float64(Float64(x * x) + Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * x) <= 1.55e-193) || (~(((x * x) <= 3.2e-155)) && ((x * x) <= 9e-57)))
		tmp = y * (y * 3.0);
	else
		tmp = (x * x) + (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 1.55e-193], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 3.2e-155]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 9e-57]]], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.5500000000000001e-193 or 3.20000000000000013e-155 < (*.f64 x x) < 8.99999999999999945e-57

   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 93.6%

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

   4. Simplified 93.6%

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

   if 1.5500000000000001e-193 < (*.f64 x x) < 3.20000000000000013e-155 or 8.99999999999999945e-57 < (*.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. add-sqr-sqrt 99.9%

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

      2. fma-def 99.9%

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

      3. +-commutative 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. hypot-def 100.0%

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

      6. hypot-def 100.0%

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

      7. +-commutative 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. hypot-def 99.9%

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

      10. hypot-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 91.3%

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

   8. Simplified 91.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.55 \cdot 10^{-193} \lor \neg \left(x \cdot x \leq 3.2 \cdot 10^{-155}\right) \land x \cdot x \leq 9 \cdot 10^{-57}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot x + y \cdot y\\ \end{array} \]

Alternative 3: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-155} \lor \neg \left(y \cdot y \leq 2 \cdot 10^{-96}\right) \land y \cdot y \leq 10^{+58}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (* y y) 2e-155) (and (not (<= (* y y) 2e-96)) (<= (* y y) 1e+58)))
   (* x x)
   (* y (* y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (((y * y) <= 2e-155) || (!((y * y) <= 2e-96) && ((y * y) <= 1e+58))) {
		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) <= 2d-155) .or. (.not. ((y * y) <= 2d-96)) .and. ((y * y) <= 1d+58)) 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) <= 2e-155) || (!((y * y) <= 2e-96) && ((y * y) <= 1e+58))) {
		tmp = x * x;
	} else {
		tmp = y * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y * y) <= 2e-155) or (not ((y * y) <= 2e-96) and ((y * y) <= 1e+58)):
		tmp = x * x
	else:
		tmp = y * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((Float64(y * y) <= 2e-155) || (!(Float64(y * y) <= 2e-96) && (Float64(y * y) <= 1e+58)))
		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) <= 2e-155) || (~(((y * y) <= 2e-96)) && ((y * y) <= 1e+58)))
		tmp = x * x;
	else
		tmp = y * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[(y * y), $MachinePrecision], 2e-155], And[N[Not[LessEqual[N[(y * y), $MachinePrecision], 2e-96]], $MachinePrecision], LessEqual[N[(y * y), $MachinePrecision], 1e+58]]], N[(x * x), $MachinePrecision], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 2.00000000000000003e-155 or 1.9999999999999998e-96 < (*.f64 y y) < 9.99999999999999944e57

   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 83.7%

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

   4. Simplified 83.7%

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

   if 2.00000000000000003e-155 < (*.f64 y y) < 1.9999999999999998e-96 or 9.99999999999999944e57 < (*.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 89.2%

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

   4. Simplified 89.3%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-155} \lor \neg \left(y \cdot y \leq 2 \cdot 10^{-96}\right) \land y \cdot y \leq 10^{+58}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 4: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-155}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{elif}\;y \cdot y \leq 10^{+58}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y y) 2e-155)
   (* x x)
   (if (<= (* y y) 2e-96)
     (* (* y y) 3.0)
     (if (<= (* y y) 1e+58) (* x x) (* y (* y 3.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 2e-155) {
		tmp = x * x;
	} else if ((y * y) <= 2e-96) {
		tmp = (y * y) * 3.0;
	} else if ((y * y) <= 1e+58) {
		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) <= 2d-155) then
        tmp = x * x
    else if ((y * y) <= 2d-96) then
        tmp = (y * y) * 3.0d0
    else if ((y * y) <= 1d+58) 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) <= 2e-155) {
		tmp = x * x;
	} else if ((y * y) <= 2e-96) {
		tmp = (y * y) * 3.0;
	} else if ((y * y) <= 1e+58) {
		tmp = x * x;
	} else {
		tmp = y * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 2e-155:
		tmp = x * x
	elif (y * y) <= 2e-96:
		tmp = (y * y) * 3.0
	elif (y * y) <= 1e+58:
		tmp = x * x
	else:
		tmp = y * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 2e-155)
		tmp = Float64(x * x);
	elseif (Float64(y * y) <= 2e-96)
		tmp = Float64(Float64(y * y) * 3.0);
	elseif (Float64(y * y) <= 1e+58)
		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) <= 2e-155)
		tmp = x * x;
	elseif ((y * y) <= 2e-96)
		tmp = (y * y) * 3.0;
	elseif ((y * y) <= 1e+58)
		tmp = x * x;
	else
		tmp = y * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 2e-155], N[(x * x), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 2e-96], N[(N[(y * y), $MachinePrecision] * 3.0), $MachinePrecision], If[LessEqual[N[(y * y), $MachinePrecision], 1e+58], N[(x * x), $MachinePrecision], N[(y * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y y) < 2.00000000000000003e-155 or 1.9999999999999998e-96 < (*.f64 y y) < 9.99999999999999944e57

   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 83.7%

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

   4. Simplified 83.7%

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

   if 2.00000000000000003e-155 < (*.f64 y y) < 1.9999999999999998e-96

   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 83.0%

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

   4. Simplified 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. distribute-rgt-out 83.0%

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

      3. metadata-eval 83.0%

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

   6. Applied egg-rr 83.0%

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

   if 9.99999999999999944e57 < (*.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 89.6%

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

   4. Simplified 89.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 2 \cdot 10^{-155}:\\ \;\;\;\;x \cdot x\\ \mathbf{elif}\;y \cdot y \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 3\\ \mathbf{elif}\;y \cdot y \leq 10^{+58}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.7× speedup

Math

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

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 99.8%

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

2. Taylor expanded in x around 0 99.8%

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

4. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 75.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.8 \cdot 10^{+294}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y y) 1.8e+294) (* x x) (* y y)))

C

double code(double x, double y) {
	double tmp;
	if ((y * y) <= 1.8e+294) {
		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) <= 1.8d+294) 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) <= 1.8e+294) {
		tmp = x * x;
	} else {
		tmp = y * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * y) <= 1.8e+294:
		tmp = x * x
	else:
		tmp = y * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * y) <= 1.8e+294)
		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) <= 1.8e+294)
		tmp = x * x;
	else
		tmp = y * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * y), $MachinePrecision], 1.8e+294], N[(x * x), $MachinePrecision], N[(y * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y y) < 1.79999999999999986e294

   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 64.4%

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

   4. Simplified 64.4%

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

   if 1.79999999999999986e294 < (*.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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. fma-def 100.0%

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

      3. +-commutative 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. hypot-def 100.0%

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

      6. hypot-def 100.0%

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

      7. +-commutative 100.0%

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

      8. add-sqr-sqrt 100.0%

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

      9. hypot-def 100.0%

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

      10. hypot-def 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 96.3%

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

   8. Simplified 96.3%

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

   9. Taylor expanded in x around 0 96.3%

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

   11. Simplified 96.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot y \leq 1.8 \cdot 10^{+294}:\\ \;\;\;\;x \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot y\\ \end{array} \]

Alternative 7: 57.3% 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 52.9%

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

4. Simplified 52.9%

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

5. Final simplification 52.9%

   \[\leadsto x \cdot x \]

Alternative 8: 1.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -3.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -3.0

Julia

function code(x, y)
	return -3.0
end

MATLAB

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

Wolfram

code[x_, y_] := -3.0

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 0 59.8%

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

4. Simplified 59.8%

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

   1. distribute-lft1-in 59.8%

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

   2. metadata-eval 59.8%

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

   3. associate-*r* 59.8%

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

   4. *-commutative 59.8%

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

   5. add-sqr-sqrt 28.0%

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

   6. associate-*r* 28.0%

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

6. Applied egg-rr 28.0%

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

7. Taylor expanded in y around -inf 0.0%

   \[\leadsto \color{blue}{-3 \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot {y}^{2}\right)} \]

9. Simplified 1.2%

   \[\leadsto \color{blue}{-3} \]

10. Final simplification 1.2%

    \[\leadsto -3 \]

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 2023240 
(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)))