Radioactive exchange between two surfaces

Percentage Accurate: 86.4% → 95.3%

Time: 3.1s

Alternatives: 4

Speedup: 1.0×

• 60.6% 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: 86.4% 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: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.1 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left({x\_m}^{2}, {x\_m}^{2}, -{y}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 2.1e+127)
   (fma (pow x_m 2.0) (pow x_m 2.0) (- (pow y 4.0)))
   (pow x_m 4.0)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 2.1e+127) {
		tmp = fma(pow(x_m, 2.0), pow(x_m, 2.0), -pow(y, 4.0));
	} else {
		tmp = pow(x_m, 4.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 2.1e+127)
		tmp = fma((x_m ^ 2.0), (x_m ^ 2.0), Float64(-(y ^ 4.0)));
	else
		tmp = x_m ^ 4.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 2.1e+127], N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[Power[x$95$m, 2.0], $MachinePrecision] + (-N[Power[y, 4.0], $MachinePrecision])), $MachinePrecision], N[Power[x$95$m, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.09999999999999992e127

   1. Initial program 91.7%

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

      1. sqr-pow 91.6%

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

      2. fma-neg 95.8%

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

      3. metadata-eval 95.8%

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

      4. metadata-eval 95.8%

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

   4. Applied egg-rr 95.8%

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

   if 2.09999999999999992e127 < x

   1. Initial program 65.0%

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

   3. Taylor expanded in x around inf 87.5%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left({x}^{2}, {x}^{2}, -{y}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;{x\_m}^{4} - {y}^{4}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= x_m 1.15e+77) (- (pow x_m 4.0) (pow y 4.0)) (pow x_m 4.0)))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1.15e+77) {
		tmp = pow(x_m, 4.0) - pow(y, 4.0);
	} else {
		tmp = pow(x_m, 4.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x_m <= 1.15d+77) then
        tmp = (x_m ** 4.0d0) - (y ** 4.0d0)
    else
        tmp = x_m ** 4.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (x_m <= 1.15e+77) {
		tmp = Math.pow(x_m, 4.0) - Math.pow(y, 4.0);
	} else {
		tmp = Math.pow(x_m, 4.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if x_m <= 1.15e+77:
		tmp = math.pow(x_m, 4.0) - math.pow(y, 4.0)
	else:
		tmp = math.pow(x_m, 4.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if (x_m <= 1.15e+77)
		tmp = Float64((x_m ^ 4.0) - (y ^ 4.0));
	else
		tmp = x_m ^ 4.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if (x_m <= 1.15e+77)
		tmp = (x_m ^ 4.0) - (y ^ 4.0);
	else
		tmp = x_m ^ 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[x$95$m, 1.15e+77], N[(N[Power[x$95$m, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[x$95$m, 4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.14999999999999997e77

   1. Initial program 93.6%

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

   if 1.14999999999999997e77 < x

   1. Initial program 63.5%

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

   3. Taylor expanded in x around inf 82.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 3 \cdot 10^{-23}:\\ \;\;\;\;{x\_m}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y)
 :precision binary64
 (if (<= (pow y 4.0) 3e-23) (pow x_m 4.0) (- (pow y 4.0))))

C

x_m = fabs(x);
double code(double x_m, double y) {
	double tmp;
	if (pow(y, 4.0) <= 3e-23) {
		tmp = pow(x_m, 4.0);
	} else {
		tmp = -pow(y, 4.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y ** 4.0d0) <= 3d-23) then
        tmp = x_m ** 4.0d0
    else
        tmp = -(y ** 4.0d0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y) {
	double tmp;
	if (Math.pow(y, 4.0) <= 3e-23) {
		tmp = Math.pow(x_m, 4.0);
	} else {
		tmp = -Math.pow(y, 4.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y):
	tmp = 0
	if math.pow(y, 4.0) <= 3e-23:
		tmp = math.pow(x_m, 4.0)
	else:
		tmp = -math.pow(y, 4.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y)
	tmp = 0.0
	if ((y ^ 4.0) <= 3e-23)
		tmp = x_m ^ 4.0;
	else
		tmp = Float64(-(y ^ 4.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y)
	tmp = 0.0;
	if ((y ^ 4.0) <= 3e-23)
		tmp = x_m ^ 4.0;
	else
		tmp = -(y ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := If[LessEqual[N[Power[y, 4.0], $MachinePrecision], 3e-23], N[Power[x$95$m, 4.0], $MachinePrecision], (-N[Power[y, 4.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 y 4) < 3.00000000000000003e-23

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 93.7%

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

   if 3.00000000000000003e-23 < (pow.f64 y 4)

   1. Initial program 74.8%

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

   3. Taylor expanded in x around 0 78.0%

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

      1. neg-mul-1 78.0%

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

   5. Simplified 78.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 3 \cdot 10^{-23}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_] := N[Power[x$95$m, 4.0], $MachinePrecision]

Derivation

1. Initial program 87.5%

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

3. Taylor expanded in x around inf 58.6%

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

4. Final simplification 58.6%

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

Reproduce

herbie shell --seed 2024044 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))