Radioactive exchange between two surfaces

Percentage Accurate: 85.5% → 94.7%

Time: 4.2s

Precision: binary64

Cost: 1092

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

Math

\[{x}^{4} - {y}^{4} \]

↓

\[\begin{array}{l} t_0 := x \cdot x + y \cdot y\\ \mathbf{if}\;x \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;t_0 \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* x x) (* y y))))
   (if (<= x 5.8e+145) (* t_0 (- (* x x) (* y y))) (* (* x x) t_0))))

C

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

↓

double code(double x, double y) {
	double t_0 = (x * x) + (y * y);
	double tmp;
	if (x <= 5.8e+145) {
		tmp = t_0 * ((x * x) - (y * y));
	} else {
		tmp = (x * x) * t_0;
	}
	return tmp;
}

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

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) + (y * y)
    if (x <= 5.8d+145) then
        tmp = t_0 * ((x * x) - (y * y))
    else
        tmp = (x * x) * t_0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y) {
	double t_0 = (x * x) + (y * y);
	double tmp;
	if (x <= 5.8e+145) {
		tmp = t_0 * ((x * x) - (y * y));
	} else {
		tmp = (x * x) * t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = (x * x) + (y * y)
	tmp = 0
	if x <= 5.8e+145:
		tmp = t_0 * ((x * x) - (y * y))
	else:
		tmp = (x * x) * t_0
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = Float64(Float64(x * x) + Float64(y * y))
	tmp = 0.0
	if (x <= 5.8e+145)
		tmp = Float64(t_0 * Float64(Float64(x * x) - Float64(y * y)));
	else
		tmp = Float64(Float64(x * x) * t_0);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = (x * x) + (y * y);
	tmp = 0.0;
	if (x <= 5.8e+145)
		tmp = t_0 * ((x * x) - (y * y));
	else
		tmp = (x * x) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.8e+145], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8000000000000001e145

   1. Initial program 90.3%

      \[{x}^{4} - {y}^{4} \]

   2. Applied egg-rr 98.3%

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

      [Start] 90.3%	\[ {x}^{4} - {y}^{4} \]
      sqr-pow [=>] 90.1%	\[ \color{blue}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}} - {y}^{4} \]
      sqr-pow [=>] 90.0%	\[ {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{\left(\frac{4}{2}\right)} \cdot {y}^{\left(\frac{4}{2}\right)}} \]
      difference-of-squares [=>] 98.3%	\[ \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]
      metadata-eval [=>] 98.3%	\[ \left({x}^{\color{blue}{2}} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      pow2 [<=] 98.3%	\[ \left(\color{blue}{x \cdot x} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      metadata-eval [=>] 98.3%	\[ \left(x \cdot x + {y}^{\color{blue}{2}}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      pow2 [<=] 98.3%	\[ \left(x \cdot x + \color{blue}{y \cdot y}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      metadata-eval [=>] 98.3%	\[ \left(x \cdot x + y \cdot y\right) \cdot \left({x}^{\color{blue}{2}} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      pow2 [<=] 98.3%	\[ \left(x \cdot x + y \cdot y\right) \cdot \left(\color{blue}{x \cdot x} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      metadata-eval [=>] 98.3%	\[ \left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - {y}^{\color{blue}{2}}\right) \]
      pow2 [<=] 98.3%	\[ \left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - \color{blue}{y \cdot y}\right) \]

   if 5.8000000000000001e145 < x

   1. Initial program 65.0%

      \[{x}^{4} - {y}^{4} \]

   2. Applied egg-rr 75.0%

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

      [Start] 65.0%	\[ {x}^{4} - {y}^{4} \]
      sqr-pow [=>] 65.0%	\[ \color{blue}{{x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}} - {y}^{4} \]
      sqr-pow [=>] 65.0%	\[ {x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)} - \color{blue}{{y}^{\left(\frac{4}{2}\right)} \cdot {y}^{\left(\frac{4}{2}\right)}} \]
      difference-of-squares [=>] 75.0%	\[ \color{blue}{\left({x}^{\left(\frac{4}{2}\right)} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right)} \]
      metadata-eval [=>] 75.0%	\[ \left({x}^{\color{blue}{2}} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      pow2 [<=] 75.0%	\[ \left(\color{blue}{x \cdot x} + {y}^{\left(\frac{4}{2}\right)}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      metadata-eval [=>] 75.0%	\[ \left(x \cdot x + {y}^{\color{blue}{2}}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      pow2 [<=] 75.0%	\[ \left(x \cdot x + \color{blue}{y \cdot y}\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      metadata-eval [=>] 75.0%	\[ \left(x \cdot x + y \cdot y\right) \cdot \left({x}^{\color{blue}{2}} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      pow2 [<=] 75.0%	\[ \left(x \cdot x + y \cdot y\right) \cdot \left(\color{blue}{x \cdot x} - {y}^{\left(\frac{4}{2}\right)}\right) \]
      metadata-eval [=>] 75.0%	\[ \left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - {y}^{\color{blue}{2}}\right) \]
      pow2 [<=] 75.0%	\[ \left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - \color{blue}{y \cdot y}\right) \]

   3. Taylor expanded in x around inf 87.5%

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

   4. Simplified 87.5%

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

      [Start] 87.5%	\[ \left(x \cdot x + y \cdot y\right) \cdot {x}^{2} \]
      unpow2 [=>] 87.5%	\[ \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.7%
Cost	1092

\[\begin{array}{l} t_0 := x \cdot x + y \cdot y\\ \mathbf{if}\;x \leq 5.8 \cdot 10^{+145}:\\ \;\;\;\;t_0 \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot t_0\\ \end{array} \]

Alternative 2
Accuracy	79.3%
Cost	1033

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+109} \lor \neg \left(y \leq 1.05 \cdot 10^{-57}\right):\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right) - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \end{array} \]

Alternative 3
Accuracy	69.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+110} \lor \neg \left(y \leq 1.22 \cdot 10^{+126}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \end{array} \]

Alternative 4
Accuracy	44.6%
Cost	777

\[\begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+109} \lor \neg \left(y \leq 1.25 \cdot 10^{+126}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 5
Accuracy	33.3%
Cost	448

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

Reproduce

herbie shell --seed 2023178 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))