Radioactive exchange between two surfaces

Percentage Accurate: 86.2% → 91.5%

Time: 2.0s

Alternatives: 3

Speedup: 1.0×

• 60.3% 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: 86.2% 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: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 8.2 \cdot 10^{+126}:\\ \;\;\;\;{x}^{4} - {y_m}^{4}\\ \mathbf{elif}\;y_m \leq 2 \cdot 10^{+142}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y_m}^{4}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 8.2e+126)
   (- (pow x 4.0) (pow y_m 4.0))
   (if (<= y_m 2e+142) (pow x 4.0) (- (pow y_m 4.0)))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.2e+126) {
		tmp = pow(x, 4.0) - pow(y_m, 4.0);
	} else if (y_m <= 2e+142) {
		tmp = pow(x, 4.0);
	} else {
		tmp = -pow(y_m, 4.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 8.2d+126) then
        tmp = (x ** 4.0d0) - (y_m ** 4.0d0)
    else if (y_m <= 2d+142) then
        tmp = x ** 4.0d0
    else
        tmp = -(y_m ** 4.0d0)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.2e+126) {
		tmp = Math.pow(x, 4.0) - Math.pow(y_m, 4.0);
	} else if (y_m <= 2e+142) {
		tmp = Math.pow(x, 4.0);
	} else {
		tmp = -Math.pow(y_m, 4.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 8.2e+126:
		tmp = math.pow(x, 4.0) - math.pow(y_m, 4.0)
	elif y_m <= 2e+142:
		tmp = math.pow(x, 4.0)
	else:
		tmp = -math.pow(y_m, 4.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 8.2e+126)
		tmp = Float64((x ^ 4.0) - (y_m ^ 4.0));
	elseif (y_m <= 2e+142)
		tmp = x ^ 4.0;
	else
		tmp = Float64(-(y_m ^ 4.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 8.2e+126)
		tmp = (x ^ 4.0) - (y_m ^ 4.0);
	elseif (y_m <= 2e+142)
		tmp = x ^ 4.0;
	else
		tmp = -(y_m ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 8.2e+126], N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y$95$m, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 2e+142], N[Power[x, 4.0], $MachinePrecision], (-N[Power[y$95$m, 4.0], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if y < 8.2000000000000001e126

   1. Initial program 90.7%

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

   if 8.2000000000000001e126 < y < 2.0000000000000001e142

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf 100.0%

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

   if 2.0000000000000001e142 < y

   1. Initial program 68.4%

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

   3. Taylor expanded in x around 0 97.4%

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

      1. neg-mul-1 97.4%

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

   5. Simplified 97.4%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+126}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+142}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;{y_m}^{4} \leq 1.26 \cdot 10^{+88}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y_m}^{4}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (pow y_m 4.0) 1.26e+88) (pow x 4.0) (- (pow y_m 4.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (pow(y_m, 4.0) <= 1.26e+88) {
		tmp = pow(x, 4.0);
	} else {
		tmp = -pow(y_m, 4.0);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m ** 4.0d0) <= 1.26d+88) then
        tmp = x ** 4.0d0
    else
        tmp = -(y_m ** 4.0d0)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (Math.pow(y_m, 4.0) <= 1.26e+88) {
		tmp = Math.pow(x, 4.0);
	} else {
		tmp = -Math.pow(y_m, 4.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if math.pow(y_m, 4.0) <= 1.26e+88:
		tmp = math.pow(x, 4.0)
	else:
		tmp = -math.pow(y_m, 4.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m ^ 4.0) <= 1.26e+88)
		tmp = x ^ 4.0;
	else
		tmp = Float64(-(y_m ^ 4.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m ^ 4.0) <= 1.26e+88)
		tmp = x ^ 4.0;
	else
		tmp = -(y_m ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[Power[y$95$m, 4.0], $MachinePrecision], 1.26e+88], N[Power[x, 4.0], $MachinePrecision], (-N[Power[y$95$m, 4.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 y 4) < 1.26e88

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 89.3%

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

   if 1.26e88 < (pow.f64 y 4)

   1. Initial program 71.8%

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

   3. Taylor expanded in x around 0 83.9%

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

      1. neg-mul-1 83.9%

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

   5. Simplified 83.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 1.26 \cdot 10^{+88}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ {x}^{4} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (pow x 4.0))

C

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

Fortran

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

Java

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

Python

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

Julia

y_m = abs(y)
function code(x, y_m)
	return x ^ 4.0
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[Power[x, 4.0], $MachinePrecision]

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in x around inf 54.3%

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

4. Final simplification 54.3%

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

Reproduce

herbie shell --seed 2024020 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))