Radioactive exchange between two surfaces

Percentage Accurate: 85.3% → 99.9%

Time: 9.2s

Alternatives: 4

Speedup: 7.4×

• 60.9% 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: 85.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. sqr-pow N/A

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

   2. sqr-pow N/A

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

   3. difference-of-squares N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. unpow2 N/A

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

   13. metadata-eval N/A

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

   14. unpow2 N/A

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

   15. difference-of-squares N/A

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

   16. lower-*.f64 N/A

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

   17. lower-+.f64 N/A

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

   18. 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

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

   2. lift-fma.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. 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. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} - {y}^{4}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-y\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(x + y\right) \cdot \left(x - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (pow x 4.0) (pow y 4.0))))
   (if (<= t_0 -2e-291)
     (* (* y (* 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 <= -2e-291) {
		tmp = (y * (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 <= -2e-291) {
		tmp = (y * (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 <= -2e-291:
		tmp = (y * (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 <= -2e-291)
		tmp = Float64(Float64(y * Float64(y * y)) * Float64(-y));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x * 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 <= -2e-291)
		tmp = (y * (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, -2e-291], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * x), $MachinePrecision] * N[(x * x), $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))) < -1.99999999999999992e-291

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. sqr-pow N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. metadata-eval N/A

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

      14. unpow2 N/A

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

      15. difference-of-squares N/A

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

      16. lower-*.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. 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

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. 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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left({y}^{\color{blue}{\left(3 + 1\right)}}\right) \]

      4. pow-plus N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -1.99999999999999992e-291 < (-.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 100.0

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

   5. Applied rewrites 100.0%

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow-prod-down N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   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. sqr-pow N/A

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

      2. sqr-pow N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. metadata-eval N/A

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

      14. unpow2 N/A

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

      15. difference-of-squares N/A

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

      16. lower-*.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. 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 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-y\right)\\ \mathbf{elif}\;{x}^{4} - {y}^{4} \leq \infty:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(x + y\right) \cdot \left(x - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -2e-291)
   (* (* y (* y y)) (- y))
   (* (* x x) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((pow(x, 4.0) - pow(y, 4.0)) <= -2e-291) {
		tmp = (y * (y * y)) * -y;
	} else {
		tmp = (x * x) * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x ** 4.0d0) - (y ** 4.0d0)) <= (-2d-291)) then
        tmp = (y * (y * y)) * -y
    else
        tmp = (x * x) * (x * x)
    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)) <= -2e-291) {
		tmp = (y * (y * y)) * -y;
	} else {
		tmp = (x * x) * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.pow(x, 4.0) - math.pow(y, 4.0)) <= -2e-291:
		tmp = (y * (y * y)) * -y
	else:
		tmp = (x * x) * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64((x ^ 4.0) - (y ^ 4.0)) <= -2e-291)
		tmp = Float64(Float64(y * Float64(y * y)) * Float64(-y));
	else
		tmp = Float64(Float64(x * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x ^ 4.0) - (y ^ 4.0)) <= -2e-291)
		tmp = (y * (y * y)) * -y;
	else
		tmp = (x * x) * (x * x);
	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], -2e-291], N[(N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -1.99999999999999992e-291

   1. Initial program 100.0%

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

      1. sqr-pow N/A

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

      2. sqr-pow N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. metadata-eval N/A

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

      14. unpow2 N/A

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

      15. difference-of-squares N/A

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

      16. lower-*.f64 N/A

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

      17. lower-+.f64 N/A

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

      18. 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

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. 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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left({y}^{\color{blue}{\left(3 + 1\right)}}\right) \]

      4. pow-plus N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if -1.99999999999999992e-291 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 77.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-pow.f64 92.2

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

   5. Applied rewrites 92.2%

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

      1. metadata-eval N/A

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

      2. pow-prod-up N/A

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

      3. pow-prod-down N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 92.0

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

   7. Applied rewrites 92.0%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -2 \cdot 10^{-291}:\\ \;\;\;\;\left(y \cdot \left(y \cdot y\right)\right) \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 85.2%

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

3. Taylor expanded in x around inf

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

   1. lower-pow.f64 60.5

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

5. Applied rewrites 60.5%

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

   1. metadata-eval N/A

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

   2. pow-prod-up N/A

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

   3. pow-prod-down N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. lower-*.f64 60.4

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

7. Applied rewrites 60.4%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))