Radioactive exchange between two surfaces

Percentage Accurate: 86.5% → 96.6%

Time: 1.9s

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: 86.5% 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: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left({y_m}^{2}, -{y_m}^{2}, {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-{y_m}^{4}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2e+153)
   (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 <= 2e+153) {
		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 <= 2e+153)
		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, 2e+153], 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 < 2e153

   1. Initial program 87.4%

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

      1. sub-neg 87.4%

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

      2. +-commutative 87.4%

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

      3. sqr-pow 87.4%

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

      4. distribute-rgt-neg-in 87.4%

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

      5. fma-def 91.3%

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

      6. metadata-eval 91.3%

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

      7. metadata-eval 91.3%

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

   3. Applied egg-rr 91.3%

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

   if 2e153 < y

   1. Initial program 56.0%

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

   2. Taylor expanded in x around 0 84.0%

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

      1. neg-mul-1 84.0%

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

   4. Simplified 84.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left({y}^{2}, -{y}^{2}, {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]

Alternative 2: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;{x}^{4} - {y_m}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y_m}^{4}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.16e+77) (- (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 <= 1.16e+77) {
		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 <= 1.16d+77) 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 <= 1.16e+77) {
		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 <= 1.16e+77:
		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 <= 1.16e+77)
		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 <= 1.16e+77)
		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, 1.16e+77], 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 < 1.1600000000000001e77

   1. Initial program 88.1%

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

   if 1.1600000000000001e77 < y

   1. Initial program 63.2%

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

   2. Taylor expanded in x around 0 84.2%

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

      1. neg-mul-1 84.2%

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

   4. Simplified 84.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]

Alternative 3: 81.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;{x}^{4} \leq 5.2 \cdot 10^{-180}:\\ \;\;\;\;-{y_m}^{4}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (pow x 4.0) 5.2e-180) (- (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) <= 5.2e-180) {
		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) <= 5.2d-180) 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) <= 5.2e-180) {
		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) <= 5.2e-180:
		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) <= 5.2e-180)
		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) <= 5.2e-180)
		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], 5.2e-180], (-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) < 5.1999999999999998e-180

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.2%

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

      1. neg-mul-1 97.2%

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

   4. Simplified 97.2%

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

   if 5.1999999999999998e-180 < (pow.f64 x 4)

   1. Initial program 70.4%

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

   2. Taylor expanded in x around inf 73.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{x}^{4} \leq 5.2 \cdot 10^{-180}:\\ \;\;\;\;-{y}^{4}\\ \mathbf{else}:\\ \;\;\;\;{x}^{4}\\ \end{array} \]

Alternative 4: 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 84.4%

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

2. Taylor expanded in x around inf 59.3%

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

3. Final simplification 59.3%

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

Reproduce

herbie shell --seed 2023325 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))