Radioactive exchange between two surfaces

Percentage Accurate: 86.5% → 96.8%

Time: 4.1s

Alternatives: 4

Speedup: 12.0×

• 60.4% 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: 96.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.28e+154)
   (* (+ (* x x) (* y y)) (- (* x x) (* y y)))
   (* (* y y) (* y (- y)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.28e+154) {
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.28d+154) then
        tmp = ((x * x) + (y * y)) * ((x * x) - (y * y))
    else
        tmp = (y * y) * (y * -y)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.28e+154) {
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.28e+154:
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y))
	else:
		tmp = (y * y) * (y * -y)
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.28e+154)
		tmp = Float64(Float64(Float64(x * x) + Float64(y * y)) * Float64(Float64(x * x) - Float64(y * y)));
	else
		tmp = Float64(Float64(y * y) * Float64(y * Float64(-y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.28e+154)
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y));
	else
		tmp = (y * y) * (y * -y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.28e+154], N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2800000000000001e154

   1. Initial program 89.7%

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

      1. sqr-pow 89.6%

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

      2. sqr-pow 89.5%

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

      3. difference-of-squares 95.3%

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

      4. metadata-eval 95.3%

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

      5. pow2 95.3%

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

      6. metadata-eval 95.3%

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

      7. pow2 95.3%

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

      8. metadata-eval 95.3%

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

      9. pow2 95.3%

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

      10. metadata-eval 95.3%

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

      11. pow2 95.3%

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

   3. Applied egg-rr 95.3%

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

   if 1.2800000000000001e154 < y

   1. Initial program 48.5%

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

   2. Taylor expanded in x around 0 78.8%

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

      1. neg-mul-1 78.8%

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

   4. Simplified 78.8%

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

      1. sqr-pow 78.8%

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

      2. metadata-eval 78.8%

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

      3. pow2 78.8%

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

      4. metadata-eval 78.8%

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

      5. pow2 78.8%

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

   6. Applied egg-rr 78.8%

      \[\leadsto -\color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{+154}:\\ \;\;\;\;\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 2: 63.2% accurate, 14.4× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+148} \lor \neg \left(x \leq 1.95 \cdot 10^{+164}\right) \land x \leq 1.4 \cdot 10^{+175}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (or (<= x 7.8e+148) (and (not (<= x 1.95e+164)) (<= x 1.4e+175)))
   (* (* y y) (* y (- y)))
   (* (* x x) (* y y))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((x <= 7.8e+148) || (!(x <= 1.95e+164) && (x <= 1.4e+175))) {
		tmp = (y * y) * (y * -y);
	} else {
		tmp = (x * x) * (y * y);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= 7.8d+148) .or. (.not. (x <= 1.95d+164)) .and. (x <= 1.4d+175)) then
        tmp = (y * y) * (y * -y)
    else
        tmp = (x * x) * (y * y)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((x <= 7.8e+148) || (!(x <= 1.95e+164) && (x <= 1.4e+175))) {
		tmp = (y * y) * (y * -y);
	} else {
		tmp = (x * x) * (y * y);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if (x <= 7.8e+148) or (not (x <= 1.95e+164) and (x <= 1.4e+175)):
		tmp = (y * y) * (y * -y)
	else:
		tmp = (x * x) * (y * y)
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if ((x <= 7.8e+148) || (!(x <= 1.95e+164) && (x <= 1.4e+175)))
		tmp = Float64(Float64(y * y) * Float64(y * Float64(-y)));
	else
		tmp = Float64(Float64(x * x) * Float64(y * y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 7.8e+148) || (~((x <= 1.95e+164)) && (x <= 1.4e+175)))
		tmp = (y * y) * (y * -y);
	else
		tmp = (x * x) * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[Or[LessEqual[x, 7.8e+148], And[N[Not[LessEqual[x, 1.95e+164]], $MachinePrecision], LessEqual[x, 1.4e+175]]], N[(N[(y * y), $MachinePrecision] * N[(y * (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.80000000000000004e148 or 1.94999999999999993e164 < x < 1.4000000000000001e175

   1. Initial program 85.7%

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

   2. Taylor expanded in x around 0 64.7%

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

      1. neg-mul-1 64.7%

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

   4. Simplified 64.7%

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

      1. sqr-pow 64.6%

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

      2. metadata-eval 64.6%

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

      3. pow2 64.6%

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

      4. metadata-eval 64.6%

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

      5. pow2 64.6%

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

   6. Applied egg-rr 64.6%

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

   if 7.80000000000000004e148 < x < 1.94999999999999993e164 or 1.4000000000000001e175 < x

   1. Initial program 75.0%

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

      1. sqr-pow 75.0%

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

      2. sqr-pow 75.0%

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

      3. difference-of-squares 81.3%

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

      4. metadata-eval 81.3%

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

      5. pow2 81.3%

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

      6. metadata-eval 81.3%

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

      7. pow2 81.3%

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

      8. metadata-eval 81.3%

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

      9. pow2 81.3%

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

      10. metadata-eval 81.3%

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

      11. pow2 81.3%

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

   3. Applied egg-rr 81.3%

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

   4. Taylor expanded in x around inf 90.6%

      \[\leadsto \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{{x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 90.6%

         \[\leadsto \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 90.6%

      \[\leadsto \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   7. Taylor expanded in x around 0 68.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot {y}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 68.8%

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

      2. unpow2 68.8%

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(y \cdot y\right)} \]

      3. *-commutative 68.8%

         \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(x \cdot x\right)} \]

   9. Simplified 68.8%

      \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(x \cdot x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8 \cdot 10^{+148} \lor \neg \left(x \leq 1.95 \cdot 10^{+164}\right) \land x \leq 1.4 \cdot 10^{+175}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 3: 82.3% accurate, 15.7× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-12) (* (* x x) (+ (* x x) (* y y))) (* (* y y) (* y (- y)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-12) {
		tmp = (x * x) * ((x * x) + (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.15d-12) then
        tmp = (x * x) * ((x * x) + (y * y))
    else
        tmp = (y * y) * (y * -y)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-12) {
		tmp = (x * x) * ((x * x) + (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.15e-12:
		tmp = (x * x) * ((x * x) + (y * y))
	else:
		tmp = (y * y) * (y * -y)
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-12)
		tmp = Float64(Float64(x * x) * Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(Float64(y * y) * Float64(y * Float64(-y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e-12)
		tmp = (x * x) * ((x * x) + (y * y));
	else
		tmp = (y * y) * (y * -y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.15e-12], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.14999999999999995e-12

   1. Initial program 89.3%

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

      1. sqr-pow 89.2%

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

      2. sqr-pow 89.1%

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

      3. difference-of-squares 94.7%

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

      4. metadata-eval 94.7%

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

      5. pow2 94.7%

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

      6. metadata-eval 94.7%

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

      7. pow2 94.7%

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

      8. metadata-eval 94.7%

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

      9. pow2 94.7%

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

      10. metadata-eval 94.7%

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

      11. pow2 94.7%

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

   3. Applied egg-rr 94.7%

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

   4. Taylor expanded in x around inf 68.6%

      \[\leadsto \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{{x}^{2}} \]
   5. Step-by-step derivation

      1. unpow2 68.6%

         \[\leadsto \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   6. Simplified 68.6%

      \[\leadsto \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   if 1.14999999999999995e-12 < y

   1. Initial program 68.3%

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

   2. Taylor expanded in x around 0 75.1%

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

      1. neg-mul-1 75.1%

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

   4. Simplified 75.1%

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

      1. sqr-pow 74.9%

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

      2. metadata-eval 74.9%

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

      3. pow2 74.9%

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

      4. metadata-eval 74.9%

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

      5. pow2 74.9%

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

   6. Applied egg-rr 74.9%

      \[\leadsto -\color{blue}{\left(y \cdot y\right) \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 4: 32.4% accurate, 29.3× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \left(x \cdot x\right) \cdot \left(y \cdot y\right) \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (* (* x x) (* y y)))

C

y = abs(y);
double code(double x, double y) {
	return (x * x) * (y * y);
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * x) * (y * y)
end function

Java

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

Python

y = abs(y)
def code(x, y):
	return (x * x) * (y * y)

Julia

y = abs(y)
function code(x, y)
	return Float64(Float64(x * x) * Float64(y * y))
end

MATLAB

y = abs(y)
function tmp = code(x, y)
	tmp = (x * x) * (y * y);
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(x * x), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.4%

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

   1. sqr-pow 84.3%

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

   2. sqr-pow 84.2%

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

   3. difference-of-squares 91.2%

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

   4. metadata-eval 91.2%

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

   5. pow2 91.2%

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

   6. metadata-eval 91.2%

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

   7. pow2 91.2%

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

   8. metadata-eval 91.2%

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

   9. pow2 91.2%

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

   10. metadata-eval 91.2%

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

   11. pow2 91.2%

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

3. Applied egg-rr 91.2%

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

4. Taylor expanded in x around inf 58.5%

   \[\leadsto \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{{x}^{2}} \]
5. Step-by-step derivation

   1. unpow2 58.5%

      \[\leadsto \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

6. Simplified 58.5%

   \[\leadsto \left(x \cdot x + y \cdot y\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

7. Taylor expanded in x around 0 36.5%

   \[\leadsto \color{blue}{{x}^{2} \cdot {y}^{2}} \]
8. Step-by-step derivation

   1. unpow2 36.5%

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

   2. unpow2 36.5%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(y \cdot y\right)} \]

   3. *-commutative 36.5%

      \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(x \cdot x\right)} \]

9. Simplified 36.5%

   \[\leadsto \color{blue}{\left(y \cdot y\right) \cdot \left(x \cdot x\right)} \]

10. Final simplification 36.5%

    \[\leadsto \left(x \cdot x\right) \cdot \left(y \cdot y\right) \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))