Radioactive exchange between two surfaces

Percentage Accurate: 85.5% → 99.9%

Time: 6.2s

Alternatives: 7

Speedup: 7.4×

• 61.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. *-lft-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 99.8

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

7. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.9

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y + x\right)\right) \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot x\right) \cdot x\right) \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -2e-318)
   (* (* (* y y) (+ y x)) (- 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)) <= -2e-318) {
		tmp = ((y * y) * (y + x)) * (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)) <= (-2d-318)) then
        tmp = ((y * y) * (y + x)) * (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)) <= -2e-318) {
		tmp = ((y * y) * (y + x)) * (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)) <= -2e-318:
		tmp = ((y * y) * (y + x)) * (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)) <= -2e-318)
		tmp = Float64(Float64(Float64(y * y) * Float64(y + x)) * 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)) <= -2e-318)
		tmp = ((y * y) * (y + x)) * (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], -2e-318], N[(N[(N[(y * y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $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))) < -2.0000024e-318

   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 99.8

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

   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 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 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 99.9

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

   9. Applied rewrites 99.9%

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

   if -2.0000024e-318 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 80.5%

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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in y around 0

       \[\leadsto \left(x - y\right) \cdot \color{blue}{\left({x}^{2} \cdot y + {x}^{3}\right)} \]
   11. 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. +-commutative N/A

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

       5. unpow2 N/A

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

       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

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

   12. 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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(y + x\right)\right) \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot x\right) \cdot x\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(-y\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot x\right) \cdot x\right) \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -2e-318)
   (* (* (- y) 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)) <= -2e-318) {
		tmp = (-y * 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)) <= (-2d-318)) then
        tmp = (-y * 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)) <= -2e-318) {
		tmp = (-y * 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)) <= -2e-318:
		tmp = (-y * 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)) <= -2e-318)
		tmp = Float64(Float64(Float64(-y) * 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)) <= -2e-318)
		tmp = (-y * 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], -2e-318], N[(N[((-y) * y), $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))) < -2.0000024e-318

   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot 1 + {x}^{2} \cdot \left(\color{blue}{0} \cdot \frac{y}{x}\right)\right) \]

      4. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot 1 + {x}^{2} \cdot \color{blue}{0}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(1 + 0\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot \color{blue}{1}\right) \]

      7. *-rgt-identity N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 1.9

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

   7. Applied rewrites 1.9%

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

   8. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 1.9

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

   10. Applied rewrites 1.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. lower-neg.f64 99.8

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

   13. Applied rewrites 99.8%

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

   if -2.0000024e-318 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 80.5%

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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in y around 0

       \[\leadsto \left(x - y\right) \cdot \color{blue}{\left({x}^{2} \cdot y + {x}^{3}\right)} \]
   11. 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. +-commutative N/A

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

       5. unpow2 N/A

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

       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

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

   12. 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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -2 \cdot 10^{-318}:\\ \;\;\;\;\left(\left(-y\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(y + x\right) \cdot x\right) \cdot x\right) \cdot \left(x - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+173}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+112}:\\ \;\;\;\;\left(\left(-y\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) (* y y))))
   (if (<= x -4.7e+173) t_0 (if (<= x 1.1e+112) (* (* (- y) y) (* y y)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * x) * (y * y);
	double tmp;
	if (x <= -4.7e+173) {
		tmp = t_0;
	} else if (x <= 1.1e+112) {
		tmp = (-y * y) * (y * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (y * y)
    if (x <= (-4.7d+173)) then
        tmp = t_0
    else if (x <= 1.1d+112) then
        tmp = (-y * y) * (y * y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * x) * (y * y);
	double tmp;
	if (x <= -4.7e+173) {
		tmp = t_0;
	} else if (x <= 1.1e+112) {
		tmp = (-y * y) * (y * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * x) * (y * y)
	tmp = 0
	if x <= -4.7e+173:
		tmp = t_0
	elif x <= 1.1e+112:
		tmp = (-y * y) * (y * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * Float64(y * y))
	tmp = 0.0
	if (x <= -4.7e+173)
		tmp = t_0;
	elseif (x <= 1.1e+112)
		tmp = Float64(Float64(Float64(-y) * y) * Float64(y * y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * x) * (y * y);
	tmp = 0.0;
	if (x <= -4.7e+173)
		tmp = t_0;
	elseif (x <= 1.1e+112)
		tmp = (-y * y) * (y * y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+173], t$95$0, If[LessEqual[x, 1.1e+112], N[(N[((-y) * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.70000000000000015e173 or 1.1e112 < x

   1. Initial program 67.2%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. lift-pow.f64 N/A

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

      5. sqr-pow N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. unpow2 N/A

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

      17. metadata-eval N/A

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

      18. unpow2 N/A

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

      19. difference-of-squares N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 N/A

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

      22. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot 1 + {x}^{2} \cdot \left(\color{blue}{0} \cdot \frac{y}{x}\right)\right) \]

      4. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot 1 + {x}^{2} \cdot \color{blue}{0}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(1 + 0\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot \color{blue}{1}\right) \]

      7. *-rgt-identity N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 90.2

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

   7. Applied rewrites 90.2%

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

   8. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 54.5

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

   10. Applied rewrites 54.5%

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

   if -4.70000000000000015e173 < x < 1.1e112

   1. Initial program 93.8%

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

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

      1. distribute-lft-in N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot 1 + {x}^{2} \cdot \left(\color{blue}{0} \cdot \frac{y}{x}\right)\right) \]

      4. mul0-lft N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot 1 + {x}^{2} \cdot \color{blue}{0}\right) \]

      5. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(1 + 0\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot \color{blue}{1}\right) \]

      7. *-rgt-identity N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 48.4

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

   7. Applied rewrites 48.4%

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

   8. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 20.0

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

   10. Applied rewrites 20.0%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. mul-1-neg N/A

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

       5. lower-neg.f64 69.0

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

   13. Applied rewrites 69.0%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+173}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+112}:\\ \;\;\;\;\left(\left(-y\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

   \[{x}^{4} - {y}^{4} \]
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 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 69.6

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

7. Applied rewrites 69.6%

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

8. Final simplification 69.6%

   \[\leadsto \left(\left(y + x\right) \cdot \left(x - y\right)\right) \cdot \left(y \cdot y\right) \]
9. Add Preprocessing

Alternative 6: 33.1% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

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

   1. distribute-lft-in N/A

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

   2. distribute-lft1-in N/A

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

   3. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot 1 + {x}^{2} \cdot \left(\color{blue}{0} \cdot \frac{y}{x}\right)\right) \]

   4. mul0-lft N/A

      \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot 1 + {x}^{2} \cdot \color{blue}{0}\right) \]

   5. distribute-lft-in N/A

      \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(1 + 0\right)\right)} \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(y, y, x \cdot x\right) \cdot \left({x}^{2} \cdot \color{blue}{1}\right) \]

   7. *-rgt-identity N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 58.4

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

7. Applied rewrites 58.4%

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

8. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 28.2

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

10. Applied rewrites 28.2%

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

11. Final simplification 28.2%

    \[\leadsto \left(x \cdot x\right) \cdot \left(y \cdot y\right) \]
12. Add Preprocessing

Alternative 7: 22.3% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-pow.f64 55.1

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

5. Applied rewrites 55.1%

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

7. Applied rewrites 20.1%

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

Reproduce

herbie shell --seed 2024288 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))