Radioactive exchange between two surfaces

Percentage Accurate: 86.6% → 95.1%

Time: 4.6s

Alternatives: 5

Speedup: 4.5×

• 59.8% 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: 86.6% 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: 95.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+185}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot x, -{y}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (if (<= x -4.7e+185)
     t_0
     (if (<= x 4.1e+114) (fma (* x x) (* x x) (- (pow y 4.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -4.7e+185) {
		tmp = t_0;
	} else if (x <= 4.1e+114) {
		tmp = fma((x * x), (x * x), -pow(y, 4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (x <= -4.7e+185)
		tmp = t_0;
	elseif (x <= 4.1e+114)
		tmp = fma(Float64(x * x), Float64(x * x), Float64(-(y ^ 4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+185], t$95$0, If[LessEqual[x, 4.1e+114], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + (-N[Power[y, 4.0], $MachinePrecision])), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.69999999999999972e185 or 4.1000000000000001e114 < x

   1. Initial program 66.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-pow.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 89.4%

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

   if -4.69999999999999972e185 < x < 4.1000000000000001e114

   1. Initial program 95.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-pow.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

Alternative 2: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -4 \cdot 10^{-313}:\\ \;\;\;\;\left(\left(\left(-y\right) \cdot y\right) \cdot y\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -4e-313)
   (* (* (* (- 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)) <= -4e-313) {
		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)) <= (-4d-313)) 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)) <= -4e-313) {
		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)) <= -4e-313:
		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)) <= -4e-313)
		tmp = Float64(Float64(Float64(Float64(-y) * y) * y) * 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)) <= -4e-313)
		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], -4e-313], N[(N[(N[((-y) * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64))) < -4.0000000000037e-313

   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

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

      3. lower-pow.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 98.9%

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

   if -4.0000000000037e-313 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 81.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-pow.f64 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.0%

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

Alternative 3: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -4 \cdot 10^{-313}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(-y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- (pow x 4.0) (pow y 4.0)) -4e-313)
   (* (* 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)) <= -4e-313) {
		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)) <= (-4d-313)) 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)) <= -4e-313) {
		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)) <= -4e-313:
		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)) <= -4e-313)
		tmp = Float64(Float64(y * y) * Float64(Float64(-y) * 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)) <= -4e-313)
		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], -4e-313], 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))) < -4.0000000000037e-313

   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

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

      3. lower-pow.f64 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 98.9%

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

   if -4.0000000000037e-313 < (-.f64 (pow.f64 x #s(literal 4 binary64)) (pow.f64 y #s(literal 4 binary64)))

   1. Initial program 81.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-pow.f64 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} - {y}^{4} \leq -4 \cdot 10^{-313}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\left(-y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+185}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, x \cdot x, \left(\left(\left(-y\right) \cdot y\right) \cdot y\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (if (<= x -4.7e+185)
     t_0
     (if (<= x 4.1e+114) (fma (* x x) (* x x) (* (* (* (- y) y) y) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -4.7e+185) {
		tmp = t_0;
	} else if (x <= 4.1e+114) {
		tmp = fma((x * x), (x * x), (((-y * y) * y) * y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (x <= -4.7e+185)
		tmp = t_0;
	elseif (x <= 4.1e+114)
		tmp = fma(Float64(x * x), Float64(x * x), Float64(Float64(Float64(Float64(-y) * y) * y) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+185], t$95$0, If[LessEqual[x, 4.1e+114], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(N[(N[((-y) * y), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.69999999999999972e185 or 4.1000000000000001e114 < x

   1. Initial program 66.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-pow.f64 89.4

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

   5. Applied rewrites 89.4%

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

   7. Applied rewrites 89.4%

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

   if -4.69999999999999972e185 < x < 4.1000000000000001e114

   1. Initial program 95.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-pow.f64 N/A

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

      4. sqr-pow N/A

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot x, -{y}^{4}\right)} \]
   5. Step-by-step derivation

      1. lift-neg.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. sqr-pow N/A

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

      4. distribute-lft-neg-in N/A

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

      5. metadata-eval N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 99.2

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

   6. Applied rewrites 99.2%

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

Alternative 5: 57.9% 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 88.3%

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

3. Taylor expanded in y around 0

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

   1. lower-pow.f64 58.6

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

5. Applied rewrites 58.6%

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

7. Applied rewrites 58.5%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))