Radioactive exchange between two surfaces

Percentage Accurate: 85.8% → 92.8%

Time: 10.5s

Alternatives: 4

Speedup: 0.7×

• 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: 85.8% 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: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 2 \cdot 10^{+287}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (pow y 4.0) 2e+287) (- (pow x 4.0) (pow y 4.0)) (- (pow y 4.0))))

C

double code(double x, double y) {
	double tmp;
	if (pow(y, 4.0) <= 2e+287) {
		tmp = pow(x, 4.0) - pow(y, 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) <= 2d+287) then
        tmp = (x ** 4.0d0) - (y ** 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) <= 2e+287) {
		tmp = Math.pow(x, 4.0) - Math.pow(y, 4.0);
	} else {
		tmp = -Math.pow(y, 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.pow(y, 4.0) <= 2e+287:
		tmp = math.pow(x, 4.0) - math.pow(y, 4.0)
	else:
		tmp = -math.pow(y, 4.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y ^ 4.0) <= 2e+287)
		tmp = Float64((x ^ 4.0) - (y ^ 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) <= 2e+287)
		tmp = (x ^ 4.0) - (y ^ 4.0);
	else
		tmp = -(y ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Power[y, 4.0], $MachinePrecision], 2e+287], N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision], (-N[Power[y, 4.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 y #s(literal 4 binary64)) < 2.0000000000000002e287

   1. Initial program 100.0%

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

   if 2.0000000000000002e287 < (pow.f64 y #s(literal 4 binary64))

   1. Initial program 70.2%

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

   3. Taylor expanded in x around 0 86.9%

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

      1. neg-mul-1 86.9%

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

   5. Simplified 86.9%

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

Alternative 2: 81.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-71} \lor \neg \left(x \leq 1.75 \cdot 10^{+31}\right):\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.75e-71) (not (<= x 1.75e+31))) (pow x 4.0) (- (pow y 4.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.75e-71) || !(x <= 1.75e+31)) {
		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 ((x <= (-1.75d-71)) .or. (.not. (x <= 1.75d+31))) 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 ((x <= -1.75e-71) || !(x <= 1.75e+31)) {
		tmp = Math.pow(x, 4.0);
	} else {
		tmp = -Math.pow(y, 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.75e-71) or not (x <= 1.75e+31):
		tmp = math.pow(x, 4.0)
	else:
		tmp = -math.pow(y, 4.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.75e-71) || !(x <= 1.75e+31))
		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 ((x <= -1.75e-71) || ~((x <= 1.75e+31)))
		tmp = x ^ 4.0;
	else
		tmp = -(y ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.75e-71], N[Not[LessEqual[x, 1.75e+31]], $MachinePrecision]], N[Power[x, 4.0], $MachinePrecision], (-N[Power[y, 4.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e-71 or 1.75e31 < x

   1. Initial program 81.1%

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

   3. Taylor expanded in x around inf 78.8%

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

   if -1.75e-71 < x < 1.75e31

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 94.8%

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

      1. neg-mul-1 94.8%

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

   5. Simplified 94.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-71} \lor \neg \left(x \leq 1.75 \cdot 10^{+31}\right):\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.3% 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 90.2%

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

3. Taylor expanded in x around inf 57.5%

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

Alternative 4: 16.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in x around 0 57.4%

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

   1. neg-mul-1 57.4%

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

5. Simplified 57.4%

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

   1. add-sqr-sqrt 14.0%

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

   2. sqrt-unprod 20.7%

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

   3. sqr-neg 20.7%

      \[\leadsto \sqrt{\color{blue}{{y}^{4} \cdot {y}^{4}}} \]

   4. sqrt-unprod 19.2%

      \[\leadsto \color{blue}{\sqrt{{y}^{4}} \cdot \sqrt{{y}^{4}}} \]

   5. add-log-exp 24.1%

      \[\leadsto \color{blue}{\log \left(e^{\sqrt{{y}^{4}} \cdot \sqrt{{y}^{4}}}\right)} \]

   6. add-sqr-sqrt 24.1%

      \[\leadsto \log \left(e^{\color{blue}{{y}^{4}}}\right) \]

   7. add-sqr-sqrt 24.1%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{{y}^{4}}} \cdot \sqrt{e^{{y}^{4}}}\right)} \]

   8. sqrt-unprod 24.1%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{{y}^{4}} \cdot e^{{y}^{4}}}\right)} \]

   9. *-un-lft-identity 24.1%

      \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot e^{\color{blue}{1 \cdot {y}^{4}}}}\right) \]

   10. exp-prod 24.1%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot \color{blue}{{\left(e^{1}\right)}^{\left({y}^{4}\right)}}}\right) \]

   11. add-sqr-sqrt 24.1%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\sqrt{{y}^{4}} \cdot \sqrt{{y}^{4}}\right)}}}\right) \]

   12. sqrt-unprod 24.1%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\sqrt{{y}^{4} \cdot {y}^{4}}\right)}}}\right) \]

   13. sqr-neg 24.1%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\left(\sqrt{\color{blue}{\left(-{y}^{4}\right) \cdot \left(-{y}^{4}\right)}}\right)}}\right) \]

   14. sqrt-unprod 14.0%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\sqrt{-{y}^{4}} \cdot \sqrt{-{y}^{4}}\right)}}}\right) \]

   15. add-sqr-sqrt 14.6%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot {\left(e^{1}\right)}^{\color{blue}{\left(-{y}^{4}\right)}}}\right) \]

   16. exp-prod 14.6%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot \color{blue}{e^{1 \cdot \left(-{y}^{4}\right)}}}\right) \]

   17. *-un-lft-identity 14.6%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot e^{\color{blue}{-{y}^{4}}}}\right) \]

   18. exp-neg 14.6%

       \[\leadsto \log \left(\sqrt{e^{{y}^{4}} \cdot \color{blue}{\frac{1}{e^{{y}^{4}}}}}\right) \]

   19. rgt-mult-inverse 15.4%

       \[\leadsto \log \left(\sqrt{\color{blue}{1}}\right) \]

7. Applied egg-rr 15.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024160 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))