Radioactive exchange between two surfaces

Percentage Accurate: 86.1% → 96.2%

Time: 4.8s

Alternatives: 6

Speedup: 12.0×

• 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: 86.1% 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: 96.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x \cdot x + y \cdot y\\ \mathbf{if}\;x \leq 1.75 \cdot 10^{+138}:\\ \;\;\;\;t_0 \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* x x) (* y y))))
   (if (<= x 1.75e+138) (* t_0 (- (* x x) (* y y))) (* (* x x) t_0))))

C

x = abs(x);
double code(double x, double y) {
	double t_0 = (x * x) + (y * y);
	double tmp;
	if (x <= 1.75e+138) {
		tmp = t_0 * ((x * x) - (y * y));
	} else {
		tmp = (x * x) * t_0;
	}
	return tmp;
}

Fortran

NOTE: x 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) :: t_0
    real(8) :: tmp
    t_0 = (x * x) + (y * y)
    if (x <= 1.75d+138) then
        tmp = t_0 * ((x * x) - (y * y))
    else
        tmp = (x * x) * t_0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y) {
	double t_0 = (x * x) + (y * y);
	double tmp;
	if (x <= 1.75e+138) {
		tmp = t_0 * ((x * x) - (y * y));
	} else {
		tmp = (x * x) * t_0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y):
	t_0 = (x * x) + (y * y)
	tmp = 0
	if x <= 1.75e+138:
		tmp = t_0 * ((x * x) - (y * y))
	else:
		tmp = (x * x) * t_0
	return tmp

Julia

x = abs(x)
function code(x, y)
	t_0 = Float64(Float64(x * x) + Float64(y * y))
	tmp = 0.0
	if (x <= 1.75e+138)
		tmp = Float64(t_0 * Float64(Float64(x * x) - Float64(y * y)));
	else
		tmp = Float64(Float64(x * x) * t_0);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y)
	t_0 = (x * x) + (y * y);
	tmp = 0.0;
	if (x <= 1.75e+138)
		tmp = t_0 * ((x * x) - (y * y));
	else
		tmp = (x * x) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.75e+138], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7499999999999999e138

   1. Initial program 86.5%

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

      1. sqr-pow 86.4%

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

      2. sqr-pow 86.2%

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

      3. difference-of-squares 96.2%

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

      4. metadata-eval 96.2%

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

      5. pow2 96.2%

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

      6. metadata-eval 96.2%

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

      7. pow2 96.2%

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

      8. metadata-eval 96.2%

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

      9. pow2 96.2%

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

      10. metadata-eval 96.2%

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

      11. pow2 96.2%

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

   3. Applied egg-rr 96.2%

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

   if 1.7499999999999999e138 < x

   1. Initial program 61.8%

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

      1. sqr-pow 61.8%

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

      2. sqr-pow 61.8%

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

      3. difference-of-squares 73.5%

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

      4. metadata-eval 73.5%

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

      5. pow2 73.5%

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

      6. metadata-eval 73.5%

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

      7. pow2 73.5%

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

      8. metadata-eval 73.5%

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

      9. pow2 73.5%

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

      10. metadata-eval 73.5%

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

      11. pow2 73.5%

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

   3. Applied egg-rr 73.5%

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

   4. Taylor expanded in x around inf 88.2%

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

      1. unpow2 88.2%

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

   6. Simplified 88.2%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{+138}:\\ \;\;\;\;\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \end{array} \]

Alternative 2: 73.8% accurate, 13.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-43}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-43)
   (* (* x x) (+ (* x x) (* y y)))
   (if (<= y 4e+148) (* (* y y) (- (* x x) (* y y))) (* (* y y) (* y (- y))))))

C

x = abs(x);
double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-43) {
		tmp = (x * x) * ((x * x) + (y * y));
	} else if (y <= 4e+148) {
		tmp = (y * y) * ((x * x) - (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Fortran

NOTE: x 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 <= 4.5d-43) then
        tmp = (x * x) * ((x * x) + (y * y))
    else if (y <= 4d+148) then
        tmp = (y * y) * ((x * x) - (y * y))
    else
        tmp = (y * y) * (y * -y)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-43) {
		tmp = (x * x) * ((x * x) + (y * y));
	} else if (y <= 4e+148) {
		tmp = (y * y) * ((x * x) - (y * y));
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y):
	tmp = 0
	if y <= 4.5e-43:
		tmp = (x * x) * ((x * x) + (y * y))
	elif y <= 4e+148:
		tmp = (y * y) * ((x * x) - (y * y))
	else:
		tmp = (y * y) * (y * -y)
	return tmp

Julia

x = abs(x)
function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-43)
		tmp = Float64(Float64(x * x) * Float64(Float64(x * x) + Float64(y * y)));
	elseif (y <= 4e+148)
		tmp = Float64(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

x = abs(x)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-43)
		tmp = (x * x) * ((x * x) + (y * y));
	elseif (y <= 4e+148)
		tmp = (y * y) * ((x * x) - (y * y));
	else
		tmp = (y * y) * (y * -y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 4.5e-43], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+148], N[(N[(y * y), $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 3 regimes

2. if y < 4.50000000000000025e-43

   1. Initial program 89.1%

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

      1. sqr-pow 89.0%

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

      2. sqr-pow 88.9%

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

      3. difference-of-squares 96.3%

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

      4. metadata-eval 96.3%

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

      5. pow2 96.3%

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

      6. metadata-eval 96.3%

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

      7. pow2 96.3%

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

      8. metadata-eval 96.3%

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

      9. pow2 96.3%

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

      10. metadata-eval 96.3%

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

      11. pow2 96.3%

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

   3. Applied egg-rr 96.3%

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

   4. Taylor expanded in x around inf 70.8%

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

      1. unpow2 70.8%

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

   6. Simplified 70.8%

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

   if 4.50000000000000025e-43 < y < 4.0000000000000002e148

   1. Initial program 79.5%

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

      1. sqr-pow 79.5%

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

      2. sqr-pow 79.3%

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

      3. difference-of-squares 99.8%

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

      4. metadata-eval 99.8%

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

      5. pow2 99.8%

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

      6. metadata-eval 99.8%

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

      7. pow2 99.8%

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

      8. metadata-eval 99.8%

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

      9. pow2 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow2 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 91.0%

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

      1. unpow2 91.0%

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

   6. Simplified 91.0%

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

   if 4.0000000000000002e148 < y

   1. Initial program 59.5%

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

      1. sqr-pow 59.5%

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

      2. sqr-pow 59.5%

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

      3. difference-of-squares 70.3%

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

      4. metadata-eval 70.3%

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

      5. pow2 70.3%

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

      6. metadata-eval 70.3%

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

      7. pow2 70.3%

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

      8. metadata-eval 70.3%

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

      9. pow2 70.3%

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

      10. metadata-eval 70.3%

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

      11. pow2 70.3%

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

   3. Applied egg-rr 70.3%

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

   4. Taylor expanded in x around 0 70.3%

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

      1. unpow2 70.3%

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

   6. Simplified 70.3%

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

   7. Taylor expanded in x around 0 81.1%

      \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 81.1%

         \[\leadsto \left(y \cdot y\right) \cdot \left(-1 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      2. mul-1-neg 81.1%

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

      3. distribute-rgt-neg-out 81.1%

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

   9. Simplified 81.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-43}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x + y \cdot y\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 3: 71.3% accurate, 15.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+141}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= x 1.02e+141) (* (* y y) (- (* x x) (* y y))) (* (* x x) (* y y))))

C

x = abs(x);
double code(double x, double y) {
	double tmp;
	if (x <= 1.02e+141) {
		tmp = (y * y) * ((x * x) - (y * y));
	} else {
		tmp = (x * x) * (y * y);
	}
	return tmp;
}

Fortran

NOTE: x 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 <= 1.02d+141) then
        tmp = (y * y) * ((x * x) - (y * y))
    else
        tmp = (x * x) * (y * y)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y) {
	double tmp;
	if (x <= 1.02e+141) {
		tmp = (y * y) * ((x * x) - (y * y));
	} else {
		tmp = (x * x) * (y * y);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y):
	tmp = 0
	if x <= 1.02e+141:
		tmp = (y * y) * ((x * x) - (y * y))
	else:
		tmp = (x * x) * (y * y)
	return tmp

Julia

x = abs(x)
function code(x, y)
	tmp = 0.0
	if (x <= 1.02e+141)
		tmp = Float64(Float64(y * y) * Float64(Float64(x * x) - Float64(y * y)));
	else
		tmp = Float64(Float64(x * x) * Float64(y * y));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.02e+141)
		tmp = (y * y) * ((x * x) - (y * y));
	else
		tmp = (x * x) * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := If[LessEqual[x, 1.02e+141], N[(N[(y * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.02e141

   1. Initial program 86.5%

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

      1. sqr-pow 86.4%

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

      2. sqr-pow 86.2%

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

      3. difference-of-squares 96.2%

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

      4. metadata-eval 96.2%

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

      5. pow2 96.2%

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

      6. metadata-eval 96.2%

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

      7. pow2 96.2%

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

      8. metadata-eval 96.2%

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

      9. pow2 96.2%

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

      10. metadata-eval 96.2%

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

      11. pow2 96.2%

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

   3. Applied egg-rr 96.2%

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

   4. Taylor expanded in x around 0 71.1%

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

      1. unpow2 71.1%

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

   6. Simplified 71.1%

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

   if 1.02e141 < x

   1. Initial program 61.8%

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

      1. sqr-pow 61.8%

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

      2. sqr-pow 61.8%

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

      3. difference-of-squares 73.5%

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

      4. metadata-eval 73.5%

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

      5. pow2 73.5%

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

      6. metadata-eval 73.5%

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

      7. pow2 73.5%

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

      8. metadata-eval 73.5%

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

      9. pow2 73.5%

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

      10. metadata-eval 73.5%

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

      11. pow2 73.5%

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

   3. Applied egg-rr 73.5%

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

   4. Taylor expanded in x around 0 47.2%

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

      1. unpow2 47.2%

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

   6. Simplified 47.2%

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

   7. Taylor expanded in y around 0 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

   9. Simplified 61.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+141}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 4: 70.1% accurate, 20.4× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+138}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= x 2.1e+138) (* (* y y) (* y (- y))) (* (* x x) (* y y))))

C

x = abs(x);
double code(double x, double y) {
	double tmp;
	if (x <= 2.1e+138) {
		tmp = (y * y) * (y * -y);
	} else {
		tmp = (x * x) * (y * y);
	}
	return tmp;
}

Fortran

NOTE: x 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 <= 2.1d+138) then
        tmp = (y * y) * (y * -y)
    else
        tmp = (x * x) * (y * y)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x, double y) {
	double tmp;
	if (x <= 2.1e+138) {
		tmp = (y * y) * (y * -y);
	} else {
		tmp = (x * x) * (y * y);
	}
	return tmp;
}

Python

x = abs(x)
def code(x, y):
	tmp = 0
	if x <= 2.1e+138:
		tmp = (y * y) * (y * -y)
	else:
		tmp = (x * x) * (y * y)
	return tmp

Julia

x = abs(x)
function code(x, y)
	tmp = 0.0
	if (x <= 2.1e+138)
		tmp = Float64(Float64(y * y) * Float64(y * Float64(-y)));
	else
		tmp = Float64(Float64(x * x) * Float64(y * y));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.1e+138)
		tmp = (y * y) * (y * -y);
	else
		tmp = (x * x) * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := If[LessEqual[x, 2.1e+138], 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 < 2.10000000000000007e138

   1. Initial program 86.5%

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

      1. sqr-pow 86.4%

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

      2. sqr-pow 86.2%

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

      3. difference-of-squares 96.2%

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

      4. metadata-eval 96.2%

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

      5. pow2 96.2%

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

      6. metadata-eval 96.2%

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

      7. pow2 96.2%

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

      8. metadata-eval 96.2%

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

      9. pow2 96.2%

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

      10. metadata-eval 96.2%

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

      11. pow2 96.2%

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

   3. Applied egg-rr 96.2%

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

   4. Taylor expanded in x around 0 71.1%

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

      1. unpow2 71.1%

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

   6. Simplified 71.1%

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

   7. Taylor expanded in x around 0 64.2%

      \[\leadsto \left(y \cdot y\right) \cdot \color{blue}{\left(-1 \cdot {y}^{2}\right)} \]
   8. Step-by-step derivation

      1. unpow2 64.2%

         \[\leadsto \left(y \cdot y\right) \cdot \left(-1 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

      2. mul-1-neg 64.2%

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

      3. distribute-rgt-neg-out 64.2%

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

   9. Simplified 64.2%

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

   if 2.10000000000000007e138 < x

   1. Initial program 61.8%

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

      1. sqr-pow 61.8%

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

      2. sqr-pow 61.8%

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

      3. difference-of-squares 73.5%

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

      4. metadata-eval 73.5%

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

      5. pow2 73.5%

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

      6. metadata-eval 73.5%

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

      7. pow2 73.5%

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

      8. metadata-eval 73.5%

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

      9. pow2 73.5%

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

      10. metadata-eval 73.5%

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

      11. pow2 73.5%

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

   3. Applied egg-rr 73.5%

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

   4. Taylor expanded in x around 0 47.2%

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

      1. unpow2 47.2%

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

   6. Simplified 47.2%

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

   7. Taylor expanded in y around 0 61.9%

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

      1. unpow2 61.9%

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

      2. unpow2 61.9%

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

   9. Simplified 61.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{+138}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 5: 32.4% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x 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

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. sqr-pow 83.1%

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

   2. sqr-pow 83.0%

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

   3. difference-of-squares 93.1%

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

   4. metadata-eval 93.1%

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

   5. pow2 93.1%

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

   6. metadata-eval 93.1%

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

   7. pow2 93.1%

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

   8. metadata-eval 93.1%

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

   9. pow2 93.1%

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

   10. metadata-eval 93.1%

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

   11. pow2 93.1%

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

3. Applied egg-rr 93.1%

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

4. Taylor expanded in x around 0 67.9%

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

   1. unpow2 67.9%

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

6. Simplified 67.9%

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

7. Taylor expanded in y around 0 32.0%

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

   1. unpow2 32.0%

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

   2. unpow2 32.0%

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

9. Simplified 32.0%

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

10. Final simplification 32.0%

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

Alternative 6: 15.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x y) :precision binary64 0.0)

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

x = Math.abs(x);
public static double code(double x, double y) {
	return 0.0;
}

Python

x = abs(x)
def code(x, y):
	return 0.0

Julia

x = abs(x)
function code(x, y)
	return 0.0
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_, y_] := 0.0

Derivation

1. Initial program 83.2%

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

   1. sqr-pow 83.1%

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

   2. sqr-pow 83.0%

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

   3. difference-of-squares 93.1%

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

   4. metadata-eval 93.1%

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

   5. pow2 93.1%

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

   6. metadata-eval 93.1%

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

   7. pow2 93.1%

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

   8. metadata-eval 93.1%

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

   9. pow2 93.1%

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

   10. metadata-eval 93.1%

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

   11. pow2 93.1%

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

3. Applied egg-rr 93.1%

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

   1. *-commutative 93.1%

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

   2. +-commutative 93.1%

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

   3. distribute-lft-in 79.5%

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

5. Applied egg-rr 79.5%

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

6. Taylor expanded in x around inf 45.5%

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

   1. unpow2 57.0%

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

8. Simplified 45.5%

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

9. Taylor expanded in x around 0 14.6%

   \[\leadsto \color{blue}{\left(-1 \cdot {y}^{2} + {y}^{2}\right) \cdot {x}^{2}} \]
10. Step-by-step derivation

    1. unpow2 14.6%

       \[\leadsto \left(-1 \cdot \color{blue}{\left(y \cdot y\right)} + {y}^{2}\right) \cdot {x}^{2} \]

    2. unpow2 14.6%

       \[\leadsto \left(-1 \cdot \left(y \cdot y\right) + \color{blue}{y \cdot y}\right) \cdot {x}^{2} \]

    3. distribute-lft1-in 14.6%

       \[\leadsto \color{blue}{\left(\left(-1 + 1\right) \cdot \left(y \cdot y\right)\right)} \cdot {x}^{2} \]

    4. metadata-eval 14.6%

       \[\leadsto \left(\color{blue}{0} \cdot \left(y \cdot y\right)\right) \cdot {x}^{2} \]

    5. mul0-lft 14.9%

       \[\leadsto \color{blue}{0} \cdot {x}^{2} \]

    6. unpow2 14.9%

       \[\leadsto 0 \cdot \color{blue}{\left(x \cdot x\right)} \]

    7. mul0-lft 15.3%

       \[\leadsto \color{blue}{0} \]

11. Simplified 15.3%

    \[\leadsto \color{blue}{0} \]

12. Final simplification 15.3%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023264 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))