Radioactive exchange between two surfaces

Percentage Accurate: 85.4% → 99.9%

Time: 5.4s

Alternatives: 5

Speedup: 7.4×

• 60.8% 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: 85.4% 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: 99.9% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. sqr-pow N/A

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval N/A

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

   13. unpow2 N/A

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

   14. distribute-lft-neg-in N/A

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

   15. lower-*.f64 N/A

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

   16. lower-neg.f64 86.6

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

4. Applied rewrites 86.6%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-pow N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lower-*.f64 86.6

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

6. Applied rewrites 86.6%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-lft-neg-out N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. cancel-sign-sub-inv N/A

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

   11. lift-*.f64 N/A

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

   12. difference-of-squares N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. difference-of-squares N/A

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

   19. lift-*.f64 N/A

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

   20. lift-fma.f64 N/A

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

8. Applied rewrites 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} - {y}^{4}\\ t_1 := \left(\left(y + x\right) \cdot \left(x - y\right)\right) \cdot \left(y \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-322}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (pow x 4.0) (pow y 4.0)))
        (t_1 (* (* (+ y x) (- x y)) (* y y))))
   (if (<= t_0 -5e-322) t_1 (if (<= t_0 INFINITY) (* (* x x) (* x x)) t_1))))

C

double code(double x, double y) {
	double t_0 = pow(x, 4.0) - pow(y, 4.0);
	double t_1 = ((y + x) * (x - y)) * (y * y);
	double tmp;
	if (t_0 <= -5e-322) {
		tmp = t_1;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x * x) * (x * x);
	} else {
		tmp = t_1;
	}
	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 t_1 = ((y + x) * (x - y)) * (y * y);
	double tmp;
	if (t_0 <= -5e-322) {
		tmp = t_1;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) * (x * x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.pow(x, 4.0) - math.pow(y, 4.0)
	t_1 = ((y + x) * (x - y)) * (y * y)
	tmp = 0
	if t_0 <= -5e-322:
		tmp = t_1
	elif t_0 <= math.inf:
		tmp = (x * x) * (x * x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64((x ^ 4.0) - (y ^ 4.0))
	t_1 = Float64(Float64(Float64(y + x) * Float64(x - y)) * Float64(y * y))
	tmp = 0.0
	if (t_0 <= -5e-322)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x * x) * Float64(x * x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x ^ 4.0) - (y ^ 4.0);
	t_1 = ((y + x) * (x - y)) * (y * y);
	tmp = 0.0;
	if (t_0 <= -5e-322)
		tmp = t_1;
	elseif (t_0 <= Inf)
		tmp = (x * x) * (x * x);
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(y + x), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-322], t$95$1, If[LessEqual[t$95$0, Infinity], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -4.99006e-322 or +inf.0 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 74.8%

      \[{x}^{4} - {y}^{4} \]
   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. +-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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 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 99.2

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

   7. Applied rewrites 99.2%

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

   if -4.99006e-322 < (-.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. 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

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(1 + \left(-1 \cdot \frac{y}{x} + \frac{y}{x}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{0} \cdot \frac{y}{x} + 1\right) \]

      6. mul0-lft N/A

         \[\leadsto {x}^{4} \cdot \left(\color{blue}{0} + 1\right) \]

      7. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

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

      9. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(y + x\right) \cdot \left(x - y\right)\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;{x}^{4} - {y}^{4} \leq \infty:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) \cdot \left(x - y\right)\right) \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot y\right) \cdot y\right) \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot x\right) \cdot x\right) \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -5e-322)
   (* (* (* (+ y x) y) y) (- x y))
   (* (* (* (+ y x) x) x) (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((pow(x, 4.0) - pow(y, 4.0)) <= -5e-322) {
		tmp = (((y + x) * y) * y) * (x - y);
	} else {
		tmp = (((y + x) * x) * x) * (x - 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 ** 4.0d0) - (y ** 4.0d0)) <= (-5d-322)) then
        tmp = (((y + x) * y) * y) * (x - y)
    else
        tmp = (((y + x) * x) * x) * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.pow(x, 4.0) - Math.pow(y, 4.0)) <= -5e-322) {
		tmp = (((y + x) * y) * y) * (x - y);
	} else {
		tmp = (((y + x) * x) * x) * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.pow(x, 4.0) - math.pow(y, 4.0)) <= -5e-322:
		tmp = (((y + x) * y) * y) * (x - y)
	else:
		tmp = (((y + x) * x) * x) * (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64((x ^ 4.0) - (y ^ 4.0)) <= -5e-322)
		tmp = Float64(Float64(Float64(Float64(y + x) * y) * y) * Float64(x - y));
	else
		tmp = Float64(Float64(Float64(Float64(y + x) * x) * x) * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x ^ 4.0) - (y ^ 4.0)) <= -5e-322)
		tmp = (((y + x) * y) * y) * (x - y);
	else
		tmp = (((y + x) * x) * x) * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], -5e-322], N[(N[(N[(N[(y + x), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y + x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x - y), $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))) < -4.99006e-322

   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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. lift-*.f64 N/A

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

      12. difference-of-squares N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. lift-*.f64 N/A

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

      20. lift-fma.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in x around 0

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

       1. cube-mult N/A

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

       2. unpow2 N/A

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

       3. distribute-rgt-out N/A

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

       4. unpow2 N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. +-commutative N/A

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

       11. lower-+.f64 99.1

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

   11. Applied rewrites 99.1%

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

   if -4.99006e-322 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 74.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 76.7

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

   4. Applied rewrites 76.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 76.7

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

   6. Applied rewrites 76.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. lift-*.f64 N/A

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

      12. difference-of-squares N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. lift-*.f64 N/A

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

      20. lift-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-out N/A

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

       4. unpow2 N/A

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

       5. +-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-+.f64 99.9

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot y\right) \cdot y\right) \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot x\right) \cdot x\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(y + x\right) \cdot \left(x - y\right)\right) \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot x\right) \cdot x\right) \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -5e-322)
   (* (* (+ y x) (- x y)) (* y y))
   (* (* (* (+ y x) x) x) (- x y))))

C

double code(double x, double y) {
	double tmp;
	if ((pow(x, 4.0) - pow(y, 4.0)) <= -5e-322) {
		tmp = ((y + x) * (x - y)) * (y * y);
	} else {
		tmp = (((y + x) * x) * x) * (x - 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 ** 4.0d0) - (y ** 4.0d0)) <= (-5d-322)) then
        tmp = ((y + x) * (x - y)) * (y * y)
    else
        tmp = (((y + x) * x) * x) * (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.pow(x, 4.0) - Math.pow(y, 4.0)) <= -5e-322) {
		tmp = ((y + x) * (x - y)) * (y * y);
	} else {
		tmp = (((y + x) * x) * x) * (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.pow(x, 4.0) - math.pow(y, 4.0)) <= -5e-322:
		tmp = ((y + x) * (x - y)) * (y * y)
	else:
		tmp = (((y + x) * x) * x) * (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64((x ^ 4.0) - (y ^ 4.0)) <= -5e-322)
		tmp = Float64(Float64(Float64(y + x) * Float64(x - y)) * Float64(y * y));
	else
		tmp = Float64(Float64(Float64(Float64(y + x) * x) * x) * Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x ^ 4.0) - (y ^ 4.0)) <= -5e-322)
		tmp = ((y + x) * (x - y)) * (y * y);
	else
		tmp = (((y + x) * x) * x) * (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], -5e-322], N[(N[(N[(y + x), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y + x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x - y), $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))) < -4.99006e-322

   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. +-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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 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 99.0

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

   7. Applied rewrites 99.0%

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

   if -4.99006e-322 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 74.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 76.7

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

   4. Applied rewrites 76.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-pow N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 76.7

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

   6. Applied rewrites 76.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-lft-neg-out N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. lift-*.f64 N/A

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

      12. difference-of-squares N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. difference-of-squares N/A

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

      19. lift-*.f64 N/A

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

      20. lift-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. distribute-lft-out N/A

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

       4. unpow2 N/A

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

       5. +-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-+.f64 99.9

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

   11. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-322}:\\ \;\;\;\;\left(\left(y + x\right) \cdot \left(x - y\right)\right) \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot x\right) \cdot x\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.5% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

   \[{x}^{4} - {y}^{4} \]
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 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

   \[\leadsto \color{blue}{{x}^{4} \cdot \left(1 + \left(-1 \cdot \frac{y}{x} + \frac{y}{x}\right)\right)} \]
6. Step-by-step derivation

   1. associate-+r+ N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

      \[\leadsto {x}^{4} \cdot \left(\color{blue}{0} \cdot \frac{y}{x} + 1\right) \]

   6. mul0-lft N/A

      \[\leadsto {x}^{4} \cdot \left(\color{blue}{0} + 1\right) \]

   7. metadata-eval N/A

      \[\leadsto {x}^{4} \cdot \color{blue}{1} \]

   8. *-rgt-identity N/A

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

   9. lower-pow.f64 53.1

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

7. Applied rewrites 53.1%

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

9. Applied rewrites 53.1%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))