Radioactive exchange between two surfaces

Percentage Accurate: 86.1% → 99.8%

Time: 4.4s

Alternatives: 3

Speedup: 1.8×

• 61.3% 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: 86.1% 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, 1.8× 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 91.8%

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

   1. sqr-pow 91.7%

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

   2. sqr-pow 91.5%

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

   3. difference-of-squares 95.5%

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

   4. metadata-eval 95.5%

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

   5. pow2 95.5%

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

   6. fma-define 95.5%

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

   7. metadata-eval 95.5%

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

   8. metadata-eval 95.5%

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

   9. metadata-eval 95.5%

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

4. Applied egg-rr 95.5%

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

   1. unpow2 95.5%

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

   2. unpow2 95.5%

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

   3. difference-of-squares 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

Alternative 2: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{x}^{4} \leq 2.9 \cdot 10^{+236}:\\ \;\;\;\;-{y}^{4}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (pow x 4.0) 2.9e+236) (- (pow y 4.0)) (pow x 4.0)))

C

double code(double x, double y) {
	double tmp;
	if (pow(x, 4.0) <= 2.9e+236) {
		tmp = -pow(y, 4.0);
	} else {
		tmp = pow(x, 4.0);
	}
	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) <= 2.9d+236) then
        tmp = -(y ** 4.0d0)
    else
        tmp = x ** 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.pow(x, 4.0) <= 2.9e+236) {
		tmp = -Math.pow(y, 4.0);
	} else {
		tmp = Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.pow(x, 4.0) <= 2.9e+236:
		tmp = -math.pow(y, 4.0)
	else:
		tmp = math.pow(x, 4.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x ^ 4.0) <= 2.9e+236)
		tmp = Float64(-(y ^ 4.0));
	else
		tmp = x ^ 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x ^ 4.0) <= 2.9e+236)
		tmp = -(y ^ 4.0);
	else
		tmp = x ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Power[x, 4.0], $MachinePrecision], 2.9e+236], (-N[Power[y, 4.0], $MachinePrecision]), N[Power[x, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 x #s(literal 4 binary64)) < 2.9000000000000001e236

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.4%

      \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]
   4. Step-by-step derivation

      1. neg-mul-1 82.4%

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

   5. Simplified 82.4%

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

   if 2.9000000000000001e236 < (pow.f64 x #s(literal 4 binary64))

   1. Initial program 78.8%

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

   3. Taylor expanded in x around inf 84.9%

      \[\leadsto \color{blue}{{x}^{4}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 57.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[Power[x, 4.0], $MachinePrecision]

Derivation

1. Initial program 91.8%

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

3. Taylor expanded in x around inf 53.4%

   \[\leadsto \color{blue}{{x}^{4}} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))