Radioactive exchange between two surfaces

Percentage Accurate: 86.1% → 99.8%

Time: 10.0s

Alternatives: 8

Speedup: 7.4×

• 61.0% 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: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip-- N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{{x}^{4} - {y}^{4}}}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{{x}^{4} - {y}^{4}}}} \]

   7. sqr-pow N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}} - {y}^{4}}} \]

   8. sqr-pow N/A

      \[\leadsto \frac{1}{\frac{1}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{\left(\frac{4}{2}\right)} \cdot {y}^{\left(\frac{4}{2}\right)}}}} \]

   9. difference-of-squares N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{1}{\left({x}^{\color{blue}{2}} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   12. unpow2 N/A

       \[\leadsto \frac{1}{\frac{1}{\left(\color{blue}{x \cdot x} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(x, x, {y}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(x, x, {y}^{\color{blue}{2}}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot y}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot y}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

4. Applied egg-rr 99.8%

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

   1. associate-*r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. --lowering--.f64 99.8

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

6. Applied egg-rr 99.8%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-lowering-+.f64 99.9

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

8. Applied egg-rr 99.9%

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

Alternative 2: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-261}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(-y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -5e-261)
   (* 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)) <= -5e-261) {
		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)) <= (-5d-261)) 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)) <= -5e-261) {
		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)) <= -5e-261:
		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)) <= -5e-261)
		tmp = Float64(y * Float64(y * Float64(y * Float64(-y))));
	else
		tmp = Float64(x * Float64(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)) <= -5e-261)
		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], -5e-261], N[(y * N[(y * N[(y * (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -4.99999999999999981e-261

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. pow-lowering-pow.f64 100.0

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

   5. Simplified 100.0%

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

      1. metadata-eval N/A

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

      2. pow-pow N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. pow3 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. *-lowering-*.f64 N/A

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

      9. cube-mult N/A

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

      10. *-lowering-*.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. neg-lowering-neg.f64 99.9

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

   7. Applied egg-rr 99.9%

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

   if -4.99999999999999981e-261 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 76.6%

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

   3. Taylor expanded in x around inf

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

      1. pow-lowering-pow.f64 89.5

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

   5. Simplified 89.5%

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

         \[\leadsto \color{blue}{{x}^{3} \cdot x} \]

      3. cube-unmult N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 89.4

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

   7. Applied egg-rr 89.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-261}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(-y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip-- N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{{x}^{4} - {y}^{4}}}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{{x}^{4} - {y}^{4}}}} \]

   7. sqr-pow N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}} - {y}^{4}}} \]

   8. sqr-pow N/A

      \[\leadsto \frac{1}{\frac{1}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{\left(\frac{4}{2}\right)} \cdot {y}^{\left(\frac{4}{2}\right)}}}} \]

   9. difference-of-squares N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{1}{\left({x}^{\color{blue}{2}} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   12. unpow2 N/A

       \[\leadsto \frac{1}{\frac{1}{\left(\color{blue}{x \cdot x} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{\mathsf{fma}\left(x, x, {y}^{\left(\frac{4}{2}\right)}\right)} \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(x, x, {y}^{\color{blue}{2}}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   15. unpow2 N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot y}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(x, x, \color{blue}{y \cdot y}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)}} \]

4. Applied egg-rr 99.8%

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

   1. associate-*r* N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. --lowering--.f64 99.8

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

6. Applied egg-rr 99.8%

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

   1. div-inv N/A

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

   2. associate-/r* N/A

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

   3. remove-double-div N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. --lowering--.f64 99.8

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

8. Applied egg-rr 99.8%

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

Alternative 4: 99.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.4%

   \[{x}^{4} - {y}^{4} \]
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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. +-lowering-+.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. --lowering--.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 egg-rr 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. *-commutative N/A

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

   2. flip3-+ N/A

      \[\leadsto \mathsf{fma}\left(x, x, y \cdot y\right) \cdot \left(\left(x - y\right) \cdot \color{blue}{\frac{{x}^{3} + {y}^{3}}{x \cdot x + \left(y \cdot y - x \cdot y\right)}}\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{fma}\left(x, x, y \cdot y\right) \cdot \left(\left(x - y\right) \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot y - x \cdot y\right)}{{x}^{3} + {y}^{3}}}}\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{fma}\left(x, x, y \cdot y\right) \cdot \left(\left(x - y\right) \cdot \frac{1}{\color{blue}{\frac{1}{\frac{{x}^{3} + {y}^{3}}{x \cdot x + \left(y \cdot y - x \cdot y\right)}}}}\right) \]

   5. flip3-+ N/A

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

   6. un-div-inv N/A

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

   7. /-lowering-/.f64 N/A

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

   8. --lowering--.f64 N/A

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

   9. /-lowering-/.f64 N/A

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

   10. +-lowering-+.f64 99.8

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

6. Applied egg-rr 99.8%

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

Alternative 5: 99.8% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.4%

   \[{x}^{4} - {y}^{4} \]
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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. +-lowering-+.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. --lowering--.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 egg-rr 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. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-lowering-*.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 6: 99.8% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.4%

   \[{x}^{4} - {y}^{4} \]
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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. +-lowering-+.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. --lowering--.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 egg-rr 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. Final simplification 99.8%

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

Alternative 7: 57.7% accurate, 12.9× speedup

Math

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

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(x * Float64(x * Float64(x * x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

3. Taylor expanded in x around inf

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

   1. pow-lowering-pow.f64 60.4

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

5. Simplified 60.4%

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

   1. metadata-eval N/A

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

   2. pow-plus N/A

      \[\leadsto \color{blue}{{x}^{3} \cdot x} \]

   3. cube-unmult N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 60.3

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

7. Applied egg-rr 60.3%

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

8. Final simplification 60.3%

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

Alternative 8: 57.7% 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 84.4%

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

3. Taylor expanded in x around inf

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

   1. pow-lowering-pow.f64 60.4

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

5. Simplified 60.4%

   \[\leadsto \color{blue}{{x}^{4}} \]
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. pow2 N/A

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

   4. pow2 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. *-lowering-*.f64 60.2

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

7. Applied egg-rr 60.2%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))