Radioactive exchange between two surfaces

Percentage Accurate: 86.3% → 95.8%

Time: 2.1s

Alternatives: 4

Speedup: 1.0×

• 60.9% 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: 86.3% 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: 95.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 2.2 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left({y_m}^{2}, -{y_m}^{2}, {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-{y_m}^{4}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2.2e+144)
   (fma (pow y_m 2.0) (- (pow y_m 2.0)) (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 <= 2.2e+144) {
		tmp = fma(pow(y_m, 2.0), -pow(y_m, 2.0), pow(x, 4.0));
	} else {
		tmp = -pow(y_m, 4.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 2.2e+144)
		tmp = fma((y_m ^ 2.0), Float64(-(y_m ^ 2.0)), (x ^ 4.0));
	else
		tmp = Float64(-(y_m ^ 4.0));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 2.2e+144], N[(N[Power[y$95$m, 2.0], $MachinePrecision] * (-N[Power[y$95$m, 2.0], $MachinePrecision]) + N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], (-N[Power[y$95$m, 4.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y < 2.19999999999999988e144

   1. Initial program 86.7%

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

      1. sub-neg 86.7%

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

      2. +-commutative 86.7%

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

      3. sqr-pow 86.6%

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

      4. distribute-rgt-neg-in 86.6%

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

      5. fma-def 90.1%

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

      6. metadata-eval 90.1%

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

      7. metadata-eval 90.1%

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

   3. Applied egg-rr 90.1%

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

   if 2.19999999999999988e144 < y

   1. Initial program 58.1%

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

   2. Taylor expanded in x around 0 90.3%

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

      1. neg-mul-1 90.3%

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

   4. Simplified 90.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left({y}^{2}, -{y}^{2}, {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]

Alternative 2: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.6 \cdot 10^{+75}:\\ \;\;\;\;{x}^{4} - {y_m}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y_m}^{4}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.6e+75) (- (pow x 4.0) (pow y_m 4.0)) (- (pow y_m 4.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.6e+75) {
		tmp = pow(x, 4.0) - pow(y_m, 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 <= 1.6d+75) then
        tmp = (x ** 4.0d0) - (y_m ** 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 <= 1.6e+75) {
		tmp = Math.pow(x, 4.0) - Math.pow(y_m, 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 <= 1.6e+75:
		tmp = math.pow(x, 4.0) - math.pow(y_m, 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 <= 1.6e+75)
		tmp = Float64((x ^ 4.0) - (y_m ^ 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 <= 1.6e+75)
		tmp = (x ^ 4.0) - (y_m ^ 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, 1.6e+75], N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y$95$m, 4.0], $MachinePrecision]), $MachinePrecision], (-N[Power[y$95$m, 4.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y < 1.59999999999999992e75

   1. Initial program 87.6%

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

   if 1.59999999999999992e75 < y

   1. Initial program 63.0%

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

   2. Taylor expanded in x around 0 89.1%

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

      1. neg-mul-1 89.1%

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

   4. Simplified 89.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{+75}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]

Alternative 3: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;{x}^{4} \leq 1.05 \cdot 10^{+31}:\\ \;\;\;\;-{y_m}^{4}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (pow x 4.0) 1.05e+31) (- (pow y_m 4.0)) (pow x 4.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (pow(x, 4.0) <= 1.05e+31) {
		tmp = -pow(y_m, 4.0);
	} else {
		tmp = pow(x, 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 ((x ** 4.0d0) <= 1.05d+31) then
        tmp = -(y_m ** 4.0d0)
    else
        tmp = x ** 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(x, 4.0) <= 1.05e+31) {
		tmp = -Math.pow(y_m, 4.0);
	} else {
		tmp = Math.pow(x, 4.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if math.pow(x, 4.0) <= 1.05e+31:
		tmp = -math.pow(y_m, 4.0)
	else:
		tmp = math.pow(x, 4.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((x ^ 4.0) <= 1.05e+31)
		tmp = Float64(-(y_m ^ 4.0));
	else
		tmp = x ^ 4.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((x ^ 4.0) <= 1.05e+31)
		tmp = -(y_m ^ 4.0);
	else
		tmp = x ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[Power[x, 4.0], $MachinePrecision], 1.05e+31], (-N[Power[y$95$m, 4.0], $MachinePrecision]), N[Power[x, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 x 4) < 1.04999999999999989e31

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 92.3%

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

      1. neg-mul-1 92.3%

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

   4. Simplified 92.3%

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

   if 1.04999999999999989e31 < (pow.f64 x 4)

   1. Initial program 67.7%

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

   2. Taylor expanded in x around inf 73.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} \leq 1.05 \cdot 10^{+31}:\\ \;\;\;\;-{y}^{4}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4}\\ \end{array} \]

Alternative 4: 57.0% 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 83.2%

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

2. Taylor expanded in x around inf 55.4%

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

3. Final simplification 55.4%

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

Reproduce

herbie shell --seed 2023319 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))