Radioactive exchange between two surfaces

Percentage Accurate: 86.4% → 95.9%

Time: 10.6s

Alternatives: 4

Speedup: 12.0×

• 61.2% 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: 86.4% 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: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+136}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1e+136)
   (* (pow (hypot x y) 2.0) (- (* x x) (* y y)))
   (* (* y y) (* y (- y)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1e+136) {
		tmp = pow(hypot(x, y), 2.0) * ((x * x) - (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+136) {
		tmp = Math.pow(Math.hypot(x, y), 2.0) * ((x * x) - (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1e+136:
		tmp = math.pow(math.hypot(x, y), 2.0) * ((x * x) - (y * y))
	else:
		tmp = (y * y) * (y * -y)
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1e+136)
		tmp = Float64((hypot(x, y) ^ 2.0) * Float64(Float64(x * x) - Float64(y * y)));
	else
		tmp = Float64(Float64(y * y) * Float64(y * Float64(-y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+136)
		tmp = (hypot(x, y) ^ 2.0) * ((x * x) - (y * y));
	else
		tmp = (y * y) * (y * -y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1e+136], N[(N[Power[N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000006e136

   1. Initial program 93.6%

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

      1. sqr-pow 93.5%

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

      2. sqr-pow 93.3%

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

      3. difference-of-squares 97.0%

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

      4. metadata-eval 97.0%

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

      5. pow2 97.0%

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

      6. metadata-eval 97.0%

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

      7. pow2 97.0%

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

      8. metadata-eval 97.0%

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

      9. pow2 97.0%

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

      10. metadata-eval 97.0%

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

      11. pow2 97.0%

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

   3. Applied egg-rr 97.0%

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

      1. sub-neg 97.0%

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

      2. distribute-lft-in 64.6%

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

      3. add-sqr-sqrt 64.6%

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

      4. pow2 64.6%

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

      5. hypot-def 64.6%

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

      6. add-sqr-sqrt 64.5%

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

      7. pow2 64.5%

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

      8. hypot-def 64.5%

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

   5. Applied egg-rr 64.5%

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

      1. distribute-lft-out 97.0%

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

      2. sub-neg 97.0%

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

   7. Simplified 97.0%

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

   if 1.00000000000000006e136 < y

   1. Initial program 59.5%

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

      1. sqr-pow 59.5%

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

      2. sqr-pow 59.5%

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

      3. difference-of-squares 67.6%

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

      4. metadata-eval 67.6%

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

      5. pow2 67.6%

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

      6. metadata-eval 67.6%

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

      7. pow2 67.6%

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

      8. metadata-eval 67.6%

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

      9. pow2 67.6%

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

      10. metadata-eval 67.6%

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

      11. pow2 67.6%

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

   3. Applied egg-rr 67.6%

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

   4. Taylor expanded in x around 0 83.8%

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

      1. unpow2 83.8%

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

      2. neg-mul-1 83.8%

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

      3. distribute-rgt-neg-in 83.8%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in x around 0 83.8%

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

      1. unpow2 83.8%

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

   9. Simplified 83.8%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+136}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 2: 96.0% accurate, 12.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.7 \cdot 10^{+138}:\\ \;\;\;\;\left(x \cdot x - y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 5.7e+138)
   (* (- (* x x) (* y y)) (+ (* x x) (* y y)))
   (* (* y y) (* y (- y)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.7e+138) {
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.7d+138) then
        tmp = ((x * x) - (y * y)) * ((x * x) + (y * y))
    else
        tmp = (y * y) * (y * -y)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.7e+138) {
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5.7e+138:
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y))
	else:
		tmp = (y * y) * (y * -y)
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5.7e+138)
		tmp = Float64(Float64(Float64(x * x) - Float64(y * y)) * Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(Float64(y * y) * Float64(y * Float64(-y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.7e+138)
		tmp = ((x * x) - (y * y)) * ((x * x) + (y * y));
	else
		tmp = (y * y) * (y * -y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 5.7e+138], N[(N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.69999999999999986e138

   1. Initial program 93.6%

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

      1. sqr-pow 93.5%

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

      2. sqr-pow 93.3%

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

      3. difference-of-squares 97.0%

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

      4. metadata-eval 97.0%

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

      5. pow2 97.0%

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

      6. metadata-eval 97.0%

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

      7. pow2 97.0%

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

      8. metadata-eval 97.0%

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

      9. pow2 97.0%

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

      10. metadata-eval 97.0%

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

      11. pow2 97.0%

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

   3. Applied egg-rr 97.0%

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

   if 5.69999999999999986e138 < y

   1. Initial program 59.5%

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

      1. sqr-pow 59.5%

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

      2. sqr-pow 59.5%

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

      3. difference-of-squares 67.6%

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

      4. metadata-eval 67.6%

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

      5. pow2 67.6%

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

      6. metadata-eval 67.6%

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

      7. pow2 67.6%

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

      8. metadata-eval 67.6%

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

      9. pow2 67.6%

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

      10. metadata-eval 67.6%

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

      11. pow2 67.6%

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

   3. Applied egg-rr 67.6%

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

   4. Taylor expanded in x around 0 83.8%

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

      1. unpow2 83.8%

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

      2. neg-mul-1 83.8%

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

      3. distribute-rgt-neg-in 83.8%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in x around 0 83.8%

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

      1. unpow2 83.8%

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

   9. Simplified 83.8%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.7 \cdot 10^{+138}:\\ \;\;\;\;\left(x \cdot x - y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 3: 69.2% accurate, 20.4× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-46}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= x 2.45e-46) (* (* y y) (* y (- y))) (* (* x x) (* x x))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (x <= 2.45e-46) {
		tmp = (y * y) * (y * -y);
	} else {
		tmp = (x * x) * (x * x);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.45d-46) then
        tmp = (y * y) * (y * -y)
    else
        tmp = (x * x) * (x * x)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (x <= 2.45e-46) {
		tmp = (y * y) * (y * -y);
	} else {
		tmp = (x * x) * (x * x);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if x <= 2.45e-46:
		tmp = (y * y) * (y * -y)
	else:
		tmp = (x * x) * (x * x)
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (x <= 2.45e-46)
		tmp = Float64(Float64(y * y) * Float64(y * Float64(-y)));
	else
		tmp = Float64(Float64(x * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.45e-46)
		tmp = (y * y) * (y * -y);
	else
		tmp = (x * x) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[x, 2.45e-46], 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 x < 2.45e-46

   1. Initial program 91.3%

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

      1. sqr-pow 91.2%

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

      2. sqr-pow 91.0%

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

      3. difference-of-squares 94.8%

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

      4. metadata-eval 94.8%

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

      5. pow2 94.8%

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

      6. metadata-eval 94.8%

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

      7. pow2 94.8%

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

      8. metadata-eval 94.8%

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

      9. pow2 94.8%

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

      10. metadata-eval 94.8%

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

      11. pow2 94.8%

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

   3. Applied egg-rr 94.8%

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

   4. Taylor expanded in x around 0 68.8%

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

      1. unpow2 68.8%

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

      2. neg-mul-1 68.8%

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

      3. distribute-rgt-neg-in 68.8%

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

   6. Simplified 68.8%

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

   7. Taylor expanded in x around 0 69.2%

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

      1. unpow2 69.2%

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

   9. Simplified 69.2%

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

   if 2.45e-46 < x

   1. Initial program 81.9%

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

      1. sqr-pow 81.7%

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

      2. sqr-pow 81.7%

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

      3. difference-of-squares 87.3%

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

      4. metadata-eval 87.3%

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

      5. pow2 87.3%

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

      6. metadata-eval 87.3%

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

      7. pow2 87.3%

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

      8. metadata-eval 87.3%

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

      9. pow2 87.3%

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

      10. metadata-eval 87.3%

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

      11. pow2 87.3%

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

   3. Applied egg-rr 87.3%

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

   4. Taylor expanded in x around inf 76.2%

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

      1. unpow2 76.2%

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

   6. Simplified 76.2%

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

   7. Taylor expanded in x around inf 76.3%

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

      1. unpow2 76.3%

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

   9. Simplified 76.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{-46}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 4: 57.3% accurate, 29.3× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (* (* x x) (* x x)))

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) * (x * x)
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	return (x * x) * (x * x);
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

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

   1. sqr-pow 88.6%

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

   2. sqr-pow 88.4%

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

   3. difference-of-squares 92.7%

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

   4. metadata-eval 92.7%

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

   5. pow2 92.7%

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

   6. metadata-eval 92.7%

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

   7. pow2 92.7%

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

   8. metadata-eval 92.7%

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

   9. pow2 92.7%

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

   10. metadata-eval 92.7%

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

   11. pow2 92.7%

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

3. Applied egg-rr 92.7%

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

4. Taylor expanded in x around inf 58.8%

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

   1. unpow2 58.8%

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

6. Simplified 58.8%

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

7. Taylor expanded in x around inf 59.1%

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

   1. unpow2 59.1%

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

9. Simplified 59.1%

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

10. Final simplification 59.1%

    \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot x\right) \]

Reproduce

herbie shell --seed 2023278 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))