Radioactive exchange between two surfaces

Percentage Accurate: 86.2% → 99.8%

Time: 3.5s

Alternatives: 4

Speedup: 1.0×

• 60.5% 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: 86.2% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x, {y}^{2}\right) \cdot \left(\left(x + y\right) \cdot \left(x - y\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (fma x x (pow y 2.0)) (* (+ x y) (- x y))))

C

double code(double x, double y) {
	return fma(x, x, pow(y, 2.0)) * ((x + y) * (x - y));
}

Julia

function code(x, y)
	return Float64(fma(x, x, (y ^ 2.0)) * Float64(Float64(x + y) * Float64(x - y)))
end

Wolfram

code[x_, y_] := N[(N[(x * x + N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

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

   1. sqr-pow 85.1%

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

   2. sqr-pow 85.0%

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

   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. fma-def 94.8%

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

   7. metadata-eval 94.8%

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

   8. metadata-eval 94.8%

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

   9. metadata-eval 94.8%

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

4. Applied egg-rr 94.8%

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

   1. unpow2 94.8%

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

   2. unpow2 94.8%

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

   3. difference-of-squares 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(x, x, {y}^{2}\right) \cdot \left(\left(x + y\right) \cdot \left(x - y\right)\right) \]
8. Add Preprocessing

Alternative 2: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+79}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+88}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e+79)
   (- (pow x 4.0) (pow y 4.0))
   (if (<= y 2.15e+88) (pow x 4.0) (- (pow y 4.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e+79) {
		tmp = pow(x, 4.0) - pow(y, 4.0);
	} else if (y <= 2.15e+88) {
		tmp = pow(x, 4.0);
	} else {
		tmp = -pow(y, 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 (y <= 4.8d+79) then
        tmp = (x ** 4.0d0) - (y ** 4.0d0)
    else if (y <= 2.15d+88) then
        tmp = x ** 4.0d0
    else
        tmp = -(y ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8e+79) {
		tmp = Math.pow(x, 4.0) - Math.pow(y, 4.0);
	} else if (y <= 2.15e+88) {
		tmp = Math.pow(x, 4.0);
	} else {
		tmp = -Math.pow(y, 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8e+79:
		tmp = math.pow(x, 4.0) - math.pow(y, 4.0)
	elif y <= 2.15e+88:
		tmp = math.pow(x, 4.0)
	else:
		tmp = -math.pow(y, 4.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e+79)
		tmp = Float64((x ^ 4.0) - (y ^ 4.0));
	elseif (y <= 2.15e+88)
		tmp = x ^ 4.0;
	else
		tmp = Float64(-(y ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e+79)
		tmp = (x ^ 4.0) - (y ^ 4.0);
	elseif (y <= 2.15e+88)
		tmp = x ^ 4.0;
	else
		tmp = -(y ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e+79], N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e+88], N[Power[x, 4.0], $MachinePrecision], (-N[Power[y, 4.0], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < 4.79999999999999971e79

   1. Initial program 89.4%

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

   if 4.79999999999999971e79 < y < 2.14999999999999987e88

   1. Initial program 50.0%

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

   3. Taylor expanded in x around inf 50.8%

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

   if 2.14999999999999987e88 < y

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0 85.1%

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

      1. neg-mul-1 85.1%

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

   5. Simplified 85.1%

      \[\leadsto \color{blue}{-{y}^{4}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+79}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{+88}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 2.8 \cdot 10^{-69}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (pow y 4.0) 2.8e-69) (pow x 4.0) (- (pow y 4.0))))

C

double code(double x, double y) {
	double tmp;
	if (pow(y, 4.0) <= 2.8e-69) {
		tmp = pow(x, 4.0);
	} else {
		tmp = -pow(y, 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 ((y ** 4.0d0) <= 2.8d-69) then
        tmp = x ** 4.0d0
    else
        tmp = -(y ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.pow(y, 4.0) <= 2.8e-69) {
		tmp = Math.pow(x, 4.0);
	} else {
		tmp = -Math.pow(y, 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.pow(y, 4.0) <= 2.8e-69:
		tmp = math.pow(x, 4.0)
	else:
		tmp = -math.pow(y, 4.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y ^ 4.0) <= 2.8e-69)
		tmp = x ^ 4.0;
	else
		tmp = Float64(-(y ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y ^ 4.0) <= 2.8e-69)
		tmp = x ^ 4.0;
	else
		tmp = -(y ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Power[y, 4.0], $MachinePrecision], 2.8e-69], N[Power[x, 4.0], $MachinePrecision], (-N[Power[y, 4.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 y 4) < 2.79999999999999979e-69

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 93.0%

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

   if 2.79999999999999979e-69 < (pow.f64 y 4)

   1. Initial program 72.0%

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

   3. Taylor expanded in x around 0 75.8%

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

      1. neg-mul-1 75.8%

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

   5. Simplified 75.8%

      \[\leadsto \color{blue}{-{y}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 2.8 \cdot 10^{-69}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.0% 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 85.2%

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

3. Taylor expanded in x around inf 56.8%

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

4. Final simplification 56.8%

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

Reproduce

herbie shell --seed 2024016 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))