Radioactive exchange between two surfaces

Percentage Accurate: 86.0% → 99.8%

Time: 7.5s

Alternatives: 9

Speedup: 7.4×

• 60.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.0% 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, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

   \[{x}^{4} - {y}^{4} \]
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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. flip3-+ N/A

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

   8. clear-num N/A

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

   9. clear-num N/A

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

   10. flip3-+ N/A

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

   11. lift-+.f64 N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. div-inv N/A

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

   10. frac-2neg N/A

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

   11. lower-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. +-commutative N/A

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

   15. distribute-neg-in N/A

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

   16. remove-double-neg N/A

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

   17. sub-neg N/A

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

   18. lower--.f64 N/A

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

   19. lift-/.f64 N/A

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

8. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. associate-/r* N/A

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

   7. div-inv N/A

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

   8. lift-/.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. lower-/.f64 N/A

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

   11. neg-mul-1 N/A

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

   12. lift-/.f64 N/A

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

   13. div-inv N/A

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

   14. lower-/.f64 99.8

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

10. Applied rewrites 99.8%

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

Alternative 2: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -2 \cdot 10^{-280}:\\ \;\;\;\;\left(\left(\left(-y\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -2e-280)
   (* (* (* (- y) y) y) y)
   (* (* (* x x) x) x)))

C

double code(double x, double y) {
	double tmp;
	if ((pow(x, 4.0) - pow(y, 4.0)) <= -2e-280) {
		tmp = ((-y * y) * y) * y;
	} else {
		tmp = ((x * x) * x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x ** 4.0d0) - (y ** 4.0d0)) <= (-2d-280)) then
        tmp = ((-y * y) * y) * y
    else
        tmp = ((x * x) * x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.pow(x, 4.0) - Math.pow(y, 4.0)) <= -2e-280) {
		tmp = ((-y * y) * y) * y;
	} else {
		tmp = ((x * x) * x) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.pow(x, 4.0) - math.pow(y, 4.0)) <= -2e-280:
		tmp = ((-y * y) * y) * y
	else:
		tmp = ((x * x) * x) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64((x ^ 4.0) - (y ^ 4.0)) <= -2e-280)
		tmp = Float64(Float64(Float64(Float64(-y) * y) * y) * y);
	else
		tmp = Float64(Float64(Float64(x * x) * x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x ^ 4.0) - (y ^ 4.0)) <= -2e-280)
		tmp = ((-y * y) * y) * y;
	else
		tmp = ((x * x) * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], -2e-280], N[(N[(N[((-y) * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -1.9999999999999999e-280

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

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

      3. lower-pow.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.3%

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

   if -1.9999999999999999e-280 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 78.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-pow.f64 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 91.4%

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

Alternative 3: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -2 \cdot 10^{-280}:\\ \;\;\;\;\left(\left(-y\right) \cdot y\right) \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -2e-280)
   (* (* (- y) y) (* y y))
   (* (* (* x x) x) x)))

C

double code(double x, double y) {
	double tmp;
	if ((pow(x, 4.0) - pow(y, 4.0)) <= -2e-280) {
		tmp = (-y * y) * (y * y);
	} else {
		tmp = ((x * x) * x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x ** 4.0d0) - (y ** 4.0d0)) <= (-2d-280)) then
        tmp = (-y * y) * (y * y)
    else
        tmp = ((x * x) * x) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.pow(x, 4.0) - Math.pow(y, 4.0)) <= -2e-280) {
		tmp = (-y * y) * (y * y);
	} else {
		tmp = ((x * x) * x) * x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.pow(x, 4.0) - math.pow(y, 4.0)) <= -2e-280:
		tmp = (-y * y) * (y * y)
	else:
		tmp = ((x * x) * x) * x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64((x ^ 4.0) - (y ^ 4.0)) <= -2e-280)
		tmp = Float64(Float64(Float64(-y) * y) * Float64(y * y));
	else
		tmp = Float64(Float64(Float64(x * x) * x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x ^ 4.0) - (y ^ 4.0)) <= -2e-280)
		tmp = (-y * y) * (y * y);
	else
		tmp = ((x * x) * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], -2e-280], N[(N[((-y) * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -1.9999999999999999e-280

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

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

      3. lower-pow.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.2%

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

   if -1.9999999999999999e-280 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 78.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-pow.f64 91.5

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

   5. Applied rewrites 91.5%

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

   7. Applied rewrites 91.4%

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

Alternative 4: 99.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

   \[{x}^{4} - {y}^{4} \]
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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. flip3-+ N/A

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

   8. clear-num N/A

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

   9. clear-num N/A

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

   10. flip3-+ N/A

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

   11. lift-+.f64 N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. div-inv N/A

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

   10. frac-2neg N/A

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

   11. lower-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. +-commutative N/A

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

   15. distribute-neg-in N/A

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

   16. remove-double-neg N/A

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

   17. sub-neg N/A

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

   18. lower--.f64 N/A

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

   19. lift-/.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 99.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

   \[{x}^{4} - {y}^{4} \]
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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. flip3-+ N/A

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

   8. clear-num N/A

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

   9. clear-num N/A

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

   10. flip3-+ N/A

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

   11. lift-+.f64 N/A

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

   12. un-div-inv N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 6: 99.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

   \[{x}^{4} - {y}^{4} \]
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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. clear-num N/A

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

   7. flip3-+ N/A

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

   8. lift-+.f64 N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 7: 99.8% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

   \[{x}^{4} - {y}^{4} \]
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. Step-by-step derivation

   1. lift-fma.f64 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 8: 56.7% accurate, 12.9× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in y around 0

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

   1. lower-pow.f64 59.3

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

5. Applied rewrites 59.3%

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

7. Applied rewrites 59.2%

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

Alternative 9: 56.7% 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 86.3%

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

3. Taylor expanded in y around 0

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

   1. lower-pow.f64 59.3

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

5. Applied rewrites 59.3%

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

7. Applied rewrites 59.1%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))