Radioactive exchange between two surfaces

Percentage Accurate: 86.7% → 95.3%

Time: 3.8s

Alternatives: 4

Speedup: 13.5×

• 59.9% 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: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {x}^{4} - {y}^{4} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (pow x 4.0) (pow y 4.0)))

C

double code(double x, double y) {
	return pow(x, 4.0) - pow(y, 4.0);
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.pow(x, 4.0) - math.pow(y, 4.0)

Julia

function code(x, y)
	return Float64((x ^ 4.0) - (y ^ 4.0))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Power[x, 4.0], $MachinePrecision] - N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+154} \lor \neg \left(x \leq 6.2 \cdot 10^{+100}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.9e+154) (not (<= x 6.2e+100)))
   (* (* x x) (* x x))
   (* (+ (* x x) (* y y)) (- (* x x) (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.9e+154) || !(x <= 6.2e+100)) {
		tmp = (x * x) * (x * x);
	} else {
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3.9d+154)) .or. (.not. (x <= 6.2d+100))) then
        tmp = (x * x) * (x * x)
    else
        tmp = ((x * x) + (y * y)) * ((x * x) - (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.9e+154) || !(x <= 6.2e+100)) {
		tmp = (x * x) * (x * x);
	} else {
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.9e+154) or not (x <= 6.2e+100):
		tmp = (x * x) * (x * x)
	else:
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.9e+154) || !(x <= 6.2e+100))
		tmp = Float64(Float64(x * x) * Float64(x * x));
	else
		tmp = Float64(Float64(Float64(x * x) + Float64(y * y)) * Float64(Float64(x * x) - Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.9e+154) || ~((x <= 6.2e+100)))
		tmp = (x * x) * (x * x);
	else
		tmp = ((x * x) + (y * y)) * ((x * x) - (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.9e+154], N[Not[LessEqual[x, 6.2e+100]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9000000000000003e154 or 6.20000000000000014e100 < x

   1. Initial program 56.5%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. sqr-pow 95.7%

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

      2. metadata-eval 95.7%

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

      3. pow2 95.7%

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

      4. metadata-eval 95.7%

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

      5. pow2 95.7%

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

   4. Applied egg-rr 95.7%

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

   if -3.9000000000000003e154 < x < 6.20000000000000014e100

   1. Initial program 95.2%

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

      1. sqr-pow 95.0%

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

      2. sqr-pow 94.8%

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

      3. difference-of-squares 99.6%

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

      4. metadata-eval 99.6%

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

      5. pow2 99.6%

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

      6. metadata-eval 99.6%

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

      7. pow2 99.6%

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

      8. metadata-eval 99.6%

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

      9. pow2 99.6%

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

      10. metadata-eval 99.6%

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

      11. pow2 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+154} \lor \neg \left(x \leq 6.2 \cdot 10^{+100}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \end{array} \]

Alternative 2: 88.2% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-66} \lor \neg \left(y \leq 1.3 \cdot 10^{-78}\right):\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -5.2e-66) (not (<= y 1.3e-78)))
   (* (* y y) (- (* x x) (* y y)))
   (* (* x x) (* x x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -5.2e-66) || !(y <= 1.3e-78)) {
		tmp = (y * y) * ((x * x) - (y * y));
	} else {
		tmp = (x * x) * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-5.2d-66)) .or. (.not. (y <= 1.3d-78))) then
        tmp = (y * y) * ((x * x) - (y * y))
    else
        tmp = (x * x) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -5.2e-66) || !(y <= 1.3e-78)) {
		tmp = (y * y) * ((x * x) - (y * y));
	} else {
		tmp = (x * x) * (x * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -5.2e-66) or not (y <= 1.3e-78):
		tmp = (y * y) * ((x * x) - (y * y))
	else:
		tmp = (x * x) * (x * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -5.2e-66) || !(y <= 1.3e-78))
		tmp = Float64(Float64(y * y) * Float64(Float64(x * x) - Float64(y * y)));
	else
		tmp = Float64(Float64(x * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -5.2e-66) || ~((y <= 1.3e-78)))
		tmp = (y * y) * ((x * x) - (y * y));
	else
		tmp = (x * x) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -5.2e-66], N[Not[LessEqual[y, 1.3e-78]], $MachinePrecision]], N[(N[(y * y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.1999999999999998e-66 or 1.3000000000000001e-78 < y

   1. Initial program 75.5%

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

      1. sqr-pow 75.5%

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

      2. sqr-pow 75.3%

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

      3. difference-of-squares 91.6%

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

      4. metadata-eval 91.6%

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

      5. pow2 91.6%

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

      6. metadata-eval 91.6%

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

      7. pow2 91.6%

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

      8. metadata-eval 91.6%

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

      9. pow2 91.6%

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

      10. metadata-eval 91.6%

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

      11. pow2 91.6%

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

   3. Applied egg-rr 91.6%

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

   4. Taylor expanded in x around 0 85.1%

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

      1. unpow2 85.1%

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

   6. Simplified 85.1%

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

   if -5.1999999999999998e-66 < y < 1.3000000000000001e-78

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. sqr-pow 99.7%

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

      2. metadata-eval 99.7%

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

      3. pow2 99.7%

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

      4. metadata-eval 99.7%

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

      5. pow2 99.7%

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-66} \lor \neg \left(y \leq 1.3 \cdot 10^{-78}\right):\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(x \cdot x - y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 3: 81.6% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-44} \lor \neg \left(x \leq 8.2 \cdot 10^{+16}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -5.3e-44) (not (<= x 8.2e+16)))
   (* (* x x) (* x x))
   (* (* y y) (* y (- y)))))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -5.3e-44) || !(x <= 8.2e+16)) {
		tmp = (x * x) * (x * x);
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-5.3d-44)) .or. (.not. (x <= 8.2d+16))) then
        tmp = (x * x) * (x * x)
    else
        tmp = (y * y) * (y * -y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -5.3e-44) || !(x <= 8.2e+16)) {
		tmp = (x * x) * (x * x);
	} else {
		tmp = (y * y) * (y * -y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -5.3e-44) or not (x <= 8.2e+16):
		tmp = (x * x) * (x * x)
	else:
		tmp = (y * y) * (y * -y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -5.3e-44) || !(x <= 8.2e+16))
		tmp = Float64(Float64(x * x) * Float64(x * x));
	else
		tmp = Float64(Float64(y * y) * Float64(y * Float64(-y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -5.3e-44) || ~((x <= 8.2e+16)))
		tmp = (x * x) * (x * x);
	else
		tmp = (y * y) * (y * -y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -5.3e-44], N[Not[LessEqual[x, 8.2e+16]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(y * y), $MachinePrecision] * N[(y * (-y)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.29999999999999971e-44 or 8.2e16 < x

   1. Initial program 69.5%

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

   2. Taylor expanded in x around inf 83.6%

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

      1. sqr-pow 83.4%

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

      2. metadata-eval 83.4%

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

      3. pow2 83.4%

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

      4. metadata-eval 83.4%

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

      5. pow2 83.4%

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

   4. Applied egg-rr 83.4%

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

   if -5.29999999999999971e-44 < x < 8.2e16

   1. Initial program 100.0%

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

      1. sqr-pow 99.9%

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

      2. sqr-pow 99.7%

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

      3. difference-of-squares 99.7%

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

      4. metadata-eval 99.7%

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

      5. pow2 99.7%

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

      6. metadata-eval 99.7%

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

      7. pow2 99.7%

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

      8. metadata-eval 99.7%

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

      9. pow2 99.7%

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

      10. metadata-eval 99.7%

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

      11. pow2 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 88.6%

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

      1. unpow2 88.6%

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

   6. Simplified 88.6%

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

   7. Taylor expanded in x around 0 88.9%

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

      1. unpow2 88.9%

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

      2. mul-1-neg 88.9%

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

      3. distribute-rgt-neg-out 88.9%

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

   9. Simplified 88.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-44} \lor \neg \left(x \leq 8.2 \cdot 10^{+16}\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(y \cdot \left(-y\right)\right)\\ \end{array} \]

Alternative 4: 57.6% accurate, 29.3× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* (* x x) (* x x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

2. Taylor expanded in x around inf 60.6%

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

   1. sqr-pow 60.4%

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

   2. metadata-eval 60.4%

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

   3. pow2 60.4%

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

   4. metadata-eval 60.4%

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

   5. pow2 60.4%

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

4. Applied egg-rr 60.4%

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

5. Final simplification 60.4%

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

Reproduce

herbie shell --seed 2023189 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))