Radioactive exchange between two surfaces

Percentage Accurate: 85.9% → 99.8%

Time: 5.4s

Alternatives: 5

Speedup: 7.4×

• 62.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ {x}^{4} - {y}^{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (pow x 4.0) (pow y 4.0)))

C

double code(double x, double y) {
	return pow(x, 4.0) - pow(y, 4.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x ** 4.0d0) - (y ** 4.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.pow(x, 4.0) - Math.pow(y, 4.0);
}

Python

def code(x, y):
	return math.pow(x, 4.0) - math.pow(y, 4.0)

Julia

function code(x, y)
	return Float64((x ^ 4.0) - (y ^ 4.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x ^ 4.0) - (y ^ 4.0);
end

Wolfram

code[x_, y_] := N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {x}^{4} - {y}^{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (pow x 4.0) (pow y 4.0)))

C

double code(double x, double y) {
	return pow(x, 4.0) - pow(y, 4.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x ** 4.0d0) - (y ** 4.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.pow(x, 4.0) - Math.pow(y, 4.0);
}

Python

def code(x, y):
	return math.pow(x, 4.0) - math.pow(y, 4.0)

Julia

function code(x, y)
	return Float64((x ^ 4.0) - (y ^ 4.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x ^ 4.0) - (y ^ 4.0);
end

Wolfram

code[x_, y_] := N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.6%

   \[{x}^{4} - {y}^{4} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{{x}^{4} - {y}^{4}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{x}^{4}} - {y}^{4} \]

   3. sqr-pow N/A

      \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}} - {y}^{4} \]

   4. lift-pow.f64 N/A

      \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{4}} \]

   5. sqr-pow N/A

      \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{\left(\frac{4}{2}\right)} \cdot {y}^{\left(\frac{4}{2}\right)}} \]

   6. difference-of-squares N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

   8. +-commutative N/A

      \[\leadsto \color{blue}{\left({y}^{\left(\frac{4}{2}\right)} + {x}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \left({y}^{\color{blue}{2}} + {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   10. unpow2 N/A

       \[\leadsto \left(\color{blue}{y \cdot y} + {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, {x}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y, y, {x}^{\color{blue}{2}}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{\color{blue}{2}} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(\color{blue}{x \cdot x} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   17. metadata-eval N/A

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

   18. unpow2 N/A

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

   19. difference-of-squares N/A

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

   20. lower-*.f64 N/A

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

   21. lower-+.f64 N/A

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

   22. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 92.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -1 \cdot 10^{-305}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(-y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -1e-305)
   (* (* y y) (* (- y) y))
   (* (fma y y (* x x)) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((pow(x, 4.0) - pow(y, 4.0)) <= -1e-305) {
		tmp = (y * y) * (-y * y);
	} else {
		tmp = fma(y, y, (x * x)) * (x * x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64((x ^ 4.0) - (y ^ 4.0)) <= -1e-305)
		tmp = Float64(Float64(y * y) * Float64(Float64(-y) * y));
	else
		tmp = Float64(fma(y, y, Float64(x * x)) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], -1e-305], N[(N[(y * y), $MachinePrecision] * N[((-y) * y), $MachinePrecision]), $MachinePrecision], N[(N[(y * y + N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -9.99999999999999996e-306

   1. Initial program 100.0%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({y}^{4}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-{y}^{4}} \]

      3. lower-pow.f64 99.4

         \[\leadsto -\color{blue}{{y}^{4}} \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{-{y}^{4}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

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

   if -9.99999999999999996e-306 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 74.2%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} - {y}^{4}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} - {y}^{4} \]

      3. sqr-pow N/A

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}} - {y}^{4} \]

      4. lift-pow.f64 N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{4}} \]

      5. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{\left(\frac{4}{2}\right)} \cdot {y}^{\left(\frac{4}{2}\right)}} \]

      6. difference-of-squares N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{\left(\frac{4}{2}\right)} + {x}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({y}^{\color{blue}{2}} + {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\color{blue}{y \cdot y} + {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, {x}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, y, {x}^{\color{blue}{2}}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{\color{blue}{2}} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(\color{blue}{x \cdot x} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      17. metadata-eval N/A

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

      18. unpow2 N/A

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

      19. difference-of-squares N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 N/A

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

      22. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(1 + \left(-1 \cdot \frac{y}{x} + \frac{y}{x}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \color{blue}{0} \cdot \frac{y}{x}\right)\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \color{blue}{0}\right)\right)\right) \]

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. lower-*.f64 86.9

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

   7. Applied rewrites 86.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -1 \cdot 10^{-305}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(-y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.0% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+136} \lor \neg \left(x \leq 5.7 \cdot 10^{+153}\right):\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(-y\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -4.4e+136) (not (<= x 5.7e+153)))
   (* (* y y) (* x x))
   (* (* y y) (* (- y) y))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -4.4e+136) || !(x <= 5.7e+153)) {
		tmp = (y * y) * (x * x);
	} else {
		tmp = (y * y) * (-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 <= (-4.4d+136)) .or. (.not. (x <= 5.7d+153))) then
        tmp = (y * y) * (x * x)
    else
        tmp = (y * y) * (-y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -4.4e+136) || !(x <= 5.7e+153)) {
		tmp = (y * y) * (x * x);
	} else {
		tmp = (y * y) * (-y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -4.4e+136) or not (x <= 5.7e+153):
		tmp = (y * y) * (x * x)
	else:
		tmp = (y * y) * (-y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -4.4e+136) || !(x <= 5.7e+153))
		tmp = Float64(Float64(y * y) * Float64(x * x));
	else
		tmp = Float64(Float64(y * y) * Float64(Float64(-y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -4.4e+136) || ~((x <= 5.7e+153)))
		tmp = (y * y) * (x * x);
	else
		tmp = (y * y) * (-y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -4.4e+136], N[Not[LessEqual[x, 5.7e+153]], $MachinePrecision]], N[(N[(y * y), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[((-y) * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999999e136 or 5.69999999999999987e153 < x

   1. Initial program 51.5%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{{x}^{4} - {y}^{4}} \]

      2. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{x}^{4}} - {y}^{4} \]

      3. sqr-pow N/A

         \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}} - {y}^{4} \]

      4. lift-pow.f64 N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{4}} \]

      5. sqr-pow N/A

         \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{\left(\frac{4}{2}\right)} \cdot {y}^{\left(\frac{4}{2}\right)}} \]

      6. difference-of-squares N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left({y}^{\left(\frac{4}{2}\right)} + {x}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \left({y}^{\color{blue}{2}} + {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(\color{blue}{y \cdot y} + {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, {x}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, y, {x}^{\color{blue}{2}}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{\color{blue}{2}} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(\color{blue}{x \cdot x} - {y}^{\left(\frac{4}{2}\right)}\right) \]

      17. metadata-eval N/A

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

      18. unpow2 N/A

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

      19. difference-of-squares N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 N/A

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

      22. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(1 + \left(-1 \cdot \frac{y}{x} + \frac{y}{x}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \color{blue}{0} \cdot \frac{y}{x}\right)\right)\right) \]

      5. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \color{blue}{0}\right)\right)\right) \]

      6. metadata-eval N/A

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

      7. *-rgt-identity N/A

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

      8. lower-*.f64 83.3

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.8

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

   10. Applied rewrites 66.8%

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

   if -4.3999999999999999e136 < x < 5.69999999999999987e153

   1. Initial program 94.7%

      \[{x}^{4} - {y}^{4} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({y}^{4}\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{-{y}^{4}} \]

      3. lower-pow.f64 77.6

         \[\leadsto -\color{blue}{{y}^{4}} \]

   5. Applied rewrites 77.6%

      \[\leadsto \color{blue}{-{y}^{4}} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+136} \lor \neg \left(x \leq 5.7 \cdot 10^{+153}\right):\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(-y\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 33.2% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

   \[{x}^{4} - {y}^{4} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{{x}^{4} - {y}^{4}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{x}^{4}} - {y}^{4} \]

   3. sqr-pow N/A

      \[\leadsto \color{blue}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}} - {y}^{4} \]

   4. lift-pow.f64 N/A

      \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{4}} \]

   5. sqr-pow N/A

      \[\leadsto {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{\left(\frac{4}{2}\right)} \cdot {y}^{\left(\frac{4}{2}\right)}} \]

   6. difference-of-squares N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]

   8. +-commutative N/A

      \[\leadsto \color{blue}{\left({y}^{\left(\frac{4}{2}\right)} + {x}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \left({y}^{\color{blue}{2}} + {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   10. unpow2 N/A

       \[\leadsto \left(\color{blue}{y \cdot y} + {x}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, y, {x}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y, y, {x}^{\color{blue}{2}}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(y, y, \color{blue}{x \cdot x}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{\color{blue}{2}} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(\color{blue}{x \cdot x} - {y}^{\left(\frac{4}{2}\right)}\right) \]

   17. metadata-eval N/A

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

   18. unpow2 N/A

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

   19. difference-of-squares N/A

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

   20. lower-*.f64 N/A

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

   21. lower-+.f64 N/A

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

   22. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(1 + \left(-1 \cdot \frac{y}{x} + \frac{y}{x}\right)\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

   3. distribute-lft1-in N/A

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

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \color{blue}{0} \cdot \frac{y}{x}\right)\right)\right) \]

   5. mul0-lft N/A

      \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(1 + \color{blue}{0}\right)\right)\right) \]

   6. metadata-eval N/A

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

   7. *-rgt-identity N/A

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

   8. lower-*.f64 55.8

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

7. Applied rewrites 55.8%

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

8. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 35.9

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

10. Applied rewrites 35.9%

    \[\leadsto \color{blue}{\left(y \cdot y\right)} \cdot \left(x \cdot x\right) \]
11. Add Preprocessing

Alternative 5: 22.0% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

   \[{x}^{4} - {y}^{4} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({y}^{4}\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto \color{blue}{-{y}^{4}} \]

   3. lower-pow.f64 62.1

      \[\leadsto -\color{blue}{{y}^{4}} \]

5. Applied rewrites 62.1%

   \[\leadsto \color{blue}{-{y}^{4}} \]
6. Step-by-step derivation

7. Applied rewrites 26.5%

   \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot y\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))