Radioactive exchange between two surfaces

Percentage Accurate: 85.9% → 99.8%

Time: 9.3s

Alternatives: 6

Speedup: 7.4×

• 61.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 89.7

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

   9. lift--.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. sqr-pow N/A

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

   12. lift-pow.f64 N/A

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

   13. sqr-pow N/A

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

   14. difference-of-squares N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 99.9

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

8. Applied rewrites 99.9%

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

Alternative 2: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(-y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -5e-285)
   (* 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-285) {
		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-285)) 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-285) {
		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-285:
		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-285)
		tmp = Float64(y * Float64(y * Float64(y * Float64(-y))));
	else
		tmp = Float64(Float64(x * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x ^ 4.0) - (y ^ 4.0)) <= -5e-285)
		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-285], N[(y * N[(y * N[(y * (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -5.00000000000000018e-285

   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. lower-neg.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 99.8%

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

   if -5.00000000000000018e-285 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 82.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-pow.f64 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 91.7%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;y \cdot \left(y \cdot \left(y \cdot \left(-y\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -5e-285)
   (* (* 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-285) {
		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-285)) 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-285) {
		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-285:
		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-285)
		tmp = Float64(Float64(y * y) * Float64(y * Float64(-y)));
	else
		tmp = Float64(Float64(x * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x ^ 4.0) - (y ^ 4.0)) <= -5e-285)
		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-285], N[(N[(y * y), $MachinePrecision] * N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -5.00000000000000018e-285

   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. lower-neg.f64 N/A

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

      3. lower-pow.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 99.8%

      \[\leadsto -\left(y \cdot y\right) \cdot \left(y \cdot y\right) \]

   if -5.00000000000000018e-285 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 82.2%

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

   3. Taylor expanded in x around inf

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

      1. lower-pow.f64 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 91.7%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -5 \cdot 10^{-285}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. 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 89.8%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. lift-pow.f64 N/A

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

   5. sqr-pow N/A

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

   6. difference-of-squares N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. unpow2 N/A

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

   16. metadata-eval N/A

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

   17. unpow2 N/A

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

   18. difference-of-squares N/A

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

   19. lower-*.f64 N/A

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

   20. lower-+.f64 N/A

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

   21. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

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

   5. clear-num N/A

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

   6. clear-num N/A

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

   7. flip3-+ N/A

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

   8. lift-+.f64 N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 5: 99.8% accurate, 7.4× speedup

Math

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

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 89.8%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. sqr-pow N/A

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

   4. lift-pow.f64 N/A

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

   5. sqr-pow N/A

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

   6. difference-of-squares N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f64 N/A

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

   11. metadata-eval N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. unpow2 N/A

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

   16. metadata-eval N/A

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

   17. unpow2 N/A

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

   18. difference-of-squares N/A

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

   19. lower-*.f64 N/A

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

   20. lower-+.f64 N/A

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

   21. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 6: 57.2% 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 89.8%

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

3. Taylor expanded in x around inf

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

   1. lower-pow.f64 53.3

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

5. Applied rewrites 53.3%

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

7. Applied rewrites 53.2%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))