Radioactive exchange between two surfaces

Percentage Accurate: 86.0% → 99.9%

Time: 4.3s

Alternatives: 5

Speedup: 1.9×

• 61.8% 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: 86.0% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \left(x + y\right)\right) \cdot \left(x - y\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* (pow (hypot x y) 2.0) (+ x y)) (- x y)))

C

double code(double x, double y) {
	return (pow(hypot(x, y), 2.0) * (x + y)) * (x - y);
}

Java

public static double code(double x, double y) {
	return (Math.pow(Math.hypot(x, y), 2.0) * (x + y)) * (x - y);
}

Python

def code(x, y):
	return (math.pow(math.hypot(x, y), 2.0) * (x + y)) * (x - y)

Julia

function code(x, y)
	return Float64(Float64((hypot(x, y) ^ 2.0) * Float64(x + y)) * Float64(x - y))
end

MATLAB

function tmp = code(x, y)
	tmp = ((hypot(x, y) ^ 2.0) * (x + y)) * (x - y);
end

Wolfram

code[x_, y_] := N[(N[(N[Power[N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision], 2.0], $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

   1. sqr-pow 82.3%

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

   2. sqr-pow 82.2%

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

   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. fma-def 91.6%

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

   7. metadata-eval 91.6%

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

   8. metadata-eval 91.6%

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

   9. metadata-eval 91.6%

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

3. Applied egg-rr 91.6%

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

   1. unpow2 91.6%

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

   2. unpow2 91.6%

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

   3. difference-of-squares 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r* 99.9%

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

   2. sub-neg 99.9%

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

   3. distribute-lft-in 71.3%

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

   4. add-sqr-sqrt 71.3%

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

   5. pow2 71.3%

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

   6. fma-udef 71.3%

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

   7. unpow2 71.3%

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

   8. hypot-def 71.3%

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

7. Applied egg-rr 71.3%

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

   1. distribute-lft-out 99.9%

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

   2. +-commutative 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

    \[\leadsto \left({\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} \cdot \left(x + y\right)\right) \cdot \left(x - y\right) \]

Alternative 2: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 2 \cdot 10^{-62} \lor \neg \left({y}^{4} \leq 10^{+61}\right) \land {y}^{4} \leq 4 \cdot 10^{+114}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot {y}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (pow y 4.0) 2e-62)
         (and (not (<= (pow y 4.0) 1e+61)) (<= (pow y 4.0) 4e+114)))
   (pow x 4.0)
   (* (- x y) (pow y 3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((pow(y, 4.0) <= 2e-62) || (!(pow(y, 4.0) <= 1e+61) && (pow(y, 4.0) <= 4e+114))) {
		tmp = pow(x, 4.0);
	} else {
		tmp = (x - y) * pow(y, 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((y ** 4.0d0) <= 2d-62) .or. (.not. ((y ** 4.0d0) <= 1d+61)) .and. ((y ** 4.0d0) <= 4d+114)) then
        tmp = x ** 4.0d0
    else
        tmp = (x - y) * (y ** 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.pow(y, 4.0) <= 2e-62) || (!(Math.pow(y, 4.0) <= 1e+61) && (Math.pow(y, 4.0) <= 4e+114))) {
		tmp = Math.pow(x, 4.0);
	} else {
		tmp = (x - y) * Math.pow(y, 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.pow(y, 4.0) <= 2e-62) or (not (math.pow(y, 4.0) <= 1e+61) and (math.pow(y, 4.0) <= 4e+114)):
		tmp = math.pow(x, 4.0)
	else:
		tmp = (x - y) * math.pow(y, 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (((y ^ 4.0) <= 2e-62) || (!((y ^ 4.0) <= 1e+61) && ((y ^ 4.0) <= 4e+114)))
		tmp = x ^ 4.0;
	else
		tmp = Float64(Float64(x - y) * (y ^ 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y ^ 4.0) <= 2e-62) || (~(((y ^ 4.0) <= 1e+61)) && ((y ^ 4.0) <= 4e+114)))
		tmp = x ^ 4.0;
	else
		tmp = (x - y) * (y ^ 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Power[y, 4.0], $MachinePrecision], 2e-62], And[N[Not[LessEqual[N[Power[y, 4.0], $MachinePrecision], 1e+61]], $MachinePrecision], LessEqual[N[Power[y, 4.0], $MachinePrecision], 4e+114]]], N[Power[x, 4.0], $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 y 4) < 2.0000000000000001e-62 or 9.99999999999999949e60 < (pow.f64 y 4) < 4e114

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 91.7%

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

   if 2.0000000000000001e-62 < (pow.f64 y 4) < 9.99999999999999949e60 or 4e114 < (pow.f64 y 4)

   1. Initial program 68.1%

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

      1. sqr-pow 68.0%

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

      2. sqr-pow 68.0%

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

      3. difference-of-squares 85.0%

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

      4. metadata-eval 85.0%

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

      5. pow2 85.0%

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

      6. fma-def 85.0%

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

      7. metadata-eval 85.0%

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

      8. metadata-eval 85.0%

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

      9. metadata-eval 85.0%

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

   3. Applied egg-rr 85.0%

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

      1. unpow2 85.0%

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

      2. unpow2 85.0%

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

      3. difference-of-squares 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*r* 99.9%

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

      2. sub-neg 99.9%

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

      3. distribute-lft-in 59.5%

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

      4. add-sqr-sqrt 59.5%

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

      5. pow2 59.5%

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

      6. fma-udef 59.5%

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

      7. unpow2 59.5%

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

      8. hypot-def 59.5%

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

   7. Applied egg-rr 59.5%

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

      1. distribute-lft-out 99.9%

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

      2. +-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around 0 87.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 2 \cdot 10^{-62} \lor \neg \left({y}^{4} \leq 10^{+61}\right) \land {y}^{4} \leq 4 \cdot 10^{+114}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot {y}^{3}\\ \end{array} \]

Alternative 3: 81.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 2.3 \cdot 10^{-51} \lor \neg \left({y}^{4} \leq 8.5 \cdot 10^{+63}\right) \land {y}^{4} \leq 3.05 \cdot 10^{+153}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= (pow y 4.0) 2.3e-51)
         (and (not (<= (pow y 4.0) 8.5e+63)) (<= (pow y 4.0) 3.05e+153)))
   (pow x 4.0)
   (- (pow y 4.0))))

C

double code(double x, double y) {
	double tmp;
	if ((pow(y, 4.0) <= 2.3e-51) || (!(pow(y, 4.0) <= 8.5e+63) && (pow(y, 4.0) <= 3.05e+153))) {
		tmp = pow(x, 4.0);
	} else {
		tmp = -pow(y, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((y ** 4.0d0) <= 2.3d-51) .or. (.not. ((y ** 4.0d0) <= 8.5d+63)) .and. ((y ** 4.0d0) <= 3.05d+153)) then
        tmp = x ** 4.0d0
    else
        tmp = -(y ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((Math.pow(y, 4.0) <= 2.3e-51) || (!(Math.pow(y, 4.0) <= 8.5e+63) && (Math.pow(y, 4.0) <= 3.05e+153))) {
		tmp = Math.pow(x, 4.0);
	} else {
		tmp = -Math.pow(y, 4.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (math.pow(y, 4.0) <= 2.3e-51) or (not (math.pow(y, 4.0) <= 8.5e+63) and (math.pow(y, 4.0) <= 3.05e+153)):
		tmp = math.pow(x, 4.0)
	else:
		tmp = -math.pow(y, 4.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (((y ^ 4.0) <= 2.3e-51) || (!((y ^ 4.0) <= 8.5e+63) && ((y ^ 4.0) <= 3.05e+153)))
		tmp = x ^ 4.0;
	else
		tmp = Float64(-(y ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y ^ 4.0) <= 2.3e-51) || (~(((y ^ 4.0) <= 8.5e+63)) && ((y ^ 4.0) <= 3.05e+153)))
		tmp = x ^ 4.0;
	else
		tmp = -(y ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[N[Power[y, 4.0], $MachinePrecision], 2.3e-51], And[N[Not[LessEqual[N[Power[y, 4.0], $MachinePrecision], 8.5e+63]], $MachinePrecision], LessEqual[N[Power[y, 4.0], $MachinePrecision], 3.05e+153]]], N[Power[x, 4.0], $MachinePrecision], (-N[Power[y, 4.0], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 y 4) < 2.30000000000000002e-51 or 8.5000000000000004e63 < (pow.f64 y 4) < 3.0499999999999999e153

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 91.8%

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

   if 2.30000000000000002e-51 < (pow.f64 y 4) < 8.5000000000000004e63 or 3.0499999999999999e153 < (pow.f64 y 4)

   1. Initial program 67.9%

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

   2. Taylor expanded in x around 0 77.9%

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

      1. neg-mul-1 77.9%

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

   4. Simplified 77.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 2.3 \cdot 10^{-51} \lor \neg \left({y}^{4} \leq 8.5 \cdot 10^{+63}\right) \land {y}^{4} \leq 3.05 \cdot 10^{+153}:\\ \;\;\;\;{x}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(x - y\right) \cdot {\left(x + y\right)}^{3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (- x y) (pow (+ x y) 3.0)))

C

double code(double x, double y) {
	return (x - y) * pow((x + y), 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x - y) * ((x + y) ** 3.0d0)
end function

Java

public static double code(double x, double y) {
	return (x - y) * Math.pow((x + y), 3.0);
}

Python

def code(x, y):
	return (x - y) * math.pow((x + y), 3.0)

Julia

function code(x, y)
	return Float64(Float64(x - y) * (Float64(x + y) ^ 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (x - y) * ((x + y) ^ 3.0);
end

Wolfram

code[x_, y_] := N[(N[(x - y), $MachinePrecision] * N[Power[N[(x + y), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.4%

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

   1. sqr-pow 82.3%

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

   2. sqr-pow 82.2%

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

   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. fma-def 91.6%

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

   7. metadata-eval 91.6%

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

   8. metadata-eval 91.6%

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

   9. metadata-eval 91.6%

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

3. Applied egg-rr 91.6%

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

   1. unpow2 91.6%

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

   2. unpow2 91.6%

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

   3. difference-of-squares 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate-*r* 99.9%

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

   2. sub-neg 99.9%

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

   3. distribute-lft-in 71.3%

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

   4. add-sqr-sqrt 71.3%

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

   5. pow2 71.3%

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

   6. fma-udef 71.3%

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

   7. unpow2 71.3%

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

   8. hypot-def 71.3%

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

7. Applied egg-rr 71.3%

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

   1. distribute-lft-out 99.9%

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

   2. +-commutative 99.9%

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

9. Simplified 99.9%

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

    1. distribute-rgt-in 76.4%

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

11. Applied egg-rr 75.7%

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

    1. +-commutative 75.7%

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

    2. distribute-rgt-out 99.1%

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

    3. pow-plus 99.2%

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

    4. metadata-eval 99.2%

       \[\leadsto {\left(x + y\right)}^{\color{blue}{3}} \cdot \left(x + \left(-y\right)\right) \]

    5. +-commutative 99.2%

       \[\leadsto {\color{blue}{\left(y + x\right)}}^{3} \cdot \left(x + \left(-y\right)\right) \]

13. Simplified 99.2%

    \[\leadsto \color{blue}{{\left(y + x\right)}^{3}} \cdot \left(x + \left(-y\right)\right) \]

14. Final simplification 99.2%

    \[\leadsto \left(x - y\right) \cdot {\left(x + y\right)}^{3} \]

Alternative 5: 57.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[Power[x, 4.0], $MachinePrecision]

Derivation

1. Initial program 82.4%

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

2. Taylor expanded in x around inf 54.2%

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

3. Final simplification 54.2%

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

Reproduce

herbie shell --seed 2023309 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))