Radioactive exchange between two surfaces

Percentage Accurate: 85.9% → 99.8%

Time: 8.4s

Alternatives: 8

Speedup: 7.4×

• 61.7% 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, 4.6× speedup

Math

\[\begin{array}{l} \\ \left(x + y\right) \cdot \frac{x - y}{\frac{1}{\mathsf{fma}\left(x, x, y \cdot y\right)}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (+ x y) (/ (- x y) (/ 1.0 (fma x x (* y y))))))

C

double code(double x, double y) {
	return (x + y) * ((x - y) / (1.0 / fma(x, x, (y * y))));
}

Julia

function code(x, y)
	return Float64(Float64(x + y) * Float64(Float64(x - y) / Float64(1.0 / fma(x, x, Float64(y * y)))))
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] * N[(N[(x - y), $MachinePrecision] / N[(1.0 / N[(x * x + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

   \[{x}^{4} - {y}^{4} \]
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. metadata-eval N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. unpow2 N/A

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

   16. metadata-eval N/A

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

   17. unpow2 N/A

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

   18. difference-of-squares N/A

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

   19. lower-*.f64 N/A

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

   20. lower-+.f64 N/A

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

   21. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. flip-+ N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. pow-pow N/A

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

   9. metadata-eval N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. pow-pow N/A

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

   14. metadata-eval N/A

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

   15. clear-num N/A

       \[\leadsto \left(x + y\right) \cdot \left(\left(x - y\right) \cdot \color{blue}{\frac{1}{\frac{x \cdot x - y \cdot y}{{x}^{4} - {y}^{4}}}}\right) \]

   16. un-div-inv N/A

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

   17. lower-/.f64 N/A

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

   18. clear-num N/A

       \[\leadsto \left(x + y\right) \cdot \frac{x - y}{\color{blue}{\frac{1}{\frac{{x}^{4} - {y}^{4}}{x \cdot x - y \cdot y}}}} \]

8. Applied rewrites 99.9%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} - {y}^{4}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-286}:\\ \;\;\;\;\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(x + y\right) \cdot \left(x - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (pow x 4.0) (pow y 4.0))))
   (if (<= t_0 -5e-286)
     (* (+ x y) (* (- x y) (* y y)))
     (if (<= t_0 INFINITY)
       (* x (* x (* x x)))
       (* (* y y) (* (+ x y) (- x y)))))))

C

double code(double x, double y) {
	double t_0 = pow(x, 4.0) - pow(y, 4.0);
	double tmp;
	if (t_0 <= -5e-286) {
		tmp = (x + y) * ((x - y) * (y * y));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = x * (x * (x * x));
	} else {
		tmp = (y * y) * ((x + y) * (x - y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.pow(x, 4.0) - Math.pow(y, 4.0);
	double tmp;
	if (t_0 <= -5e-286) {
		tmp = (x + y) * ((x - y) * (y * y));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = x * (x * (x * x));
	} else {
		tmp = (y * y) * ((x + y) * (x - y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(x, 4.0) - math.pow(y, 4.0)
	tmp = 0
	if t_0 <= -5e-286:
		tmp = (x + y) * ((x - y) * (y * y))
	elif t_0 <= math.inf:
		tmp = x * (x * (x * x))
	else:
		tmp = (y * y) * ((x + y) * (x - y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64((x ^ 4.0) - (y ^ 4.0))
	tmp = 0.0
	if (t_0 <= -5e-286)
		tmp = Float64(Float64(x + y) * Float64(Float64(x - y) * Float64(y * y)));
	elseif (t_0 <= Inf)
		tmp = Float64(x * Float64(x * Float64(x * x)));
	else
		tmp = Float64(Float64(y * y) * Float64(Float64(x + y) * Float64(x - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x ^ 4.0) - (y ^ 4.0);
	tmp = 0.0;
	if (t_0 <= -5e-286)
		tmp = (x + y) * ((x - y) * (y * y));
	elseif (t_0 <= Inf)
		tmp = x * (x * (x * x));
	else
		tmp = (y * y) * ((x + y) * (x - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-286], N[(N[(x + y), $MachinePrecision] * N[(N[(x - y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -5.00000000000000037e-286

   1. Initial program 100.0%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. unpow2 N/A

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

      18. difference-of-squares N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if -5.00000000000000037e-286 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-pow.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.1%

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

   if +inf.0 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 0.0%

      \[{x}^{4} - {y}^{4} \]
   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. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow2 N/A

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

      16. metadata-eval N/A

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

      17. unpow2 N/A

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

      18. difference-of-squares N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-286}:\\ \;\;\;\;\left(x + y\right) \cdot \left(\left(x - y\right) \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;{x}^{4} - {y}^{4} \leq \infty:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(x + y\right) \cdot \left(x - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))