Average Error: 8.5 → 4.4
Time: 1.3s
Precision: binary64
\[{x}^{4} - {y}^{4} \]
↓
\[\begin{array}{l} \mathbf{if}\;{y}^{4} \leq 1.415955287042566 \cdot 10^{+284}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]
{x}^{4} - {y}^{4}
↓
\begin{array}{l}
\mathbf{if}\;{y}^{4} \leq 1.415955287042566 \cdot 10^{+284}:\\
\;\;\;\;{x}^{4} - {y}^{4}\\

\mathbf{else}:\\
\;\;\;\;-{y}^{4}\\


\end{array}

(FPCore (x y) :precision binary64 (- (pow x 4.0) (pow y 4.0)))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (pow y 4.0) 1.415955287042566e+284)
   (- (pow x 4.0) (pow y 4.0))
   (- (pow y 4.0))))
double code(double x, double y) {
	return pow(x, 4.0) - pow(y, 4.0);
}
↓
double code(double x, double y) {
	double tmp;
	if (pow(y, 4.0) <= 1.415955287042566e+284) {
		tmp = pow(x, 4.0) - pow(y, 4.0);
	} else {
		tmp = -pow(y, 4.0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (pow.f64 y 4) < 1.41595528704256602e284

    1. Initial program 0.0

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

    if 1.41595528704256602e284 < (pow.f64 y 4)

    1. Initial program 22.0

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

    2. Taylor expanded in x around 0 11.2

      \[\leadsto \color{blue}{-1 \cdot {y}^{4}} \]

    3. Simplified11.2

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

  4. Final simplification4.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;{y}^{4} \leq 1.415955287042566 \cdot 10^{+284}:\\ \;\;\;\;{x}^{4} - {y}^{4}\\ \mathbf{else}:\\ \;\;\;\;-{y}^{4}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x y)
  :name "Radioactive exchange between two surfaces"
  :precision binary64
  (- (pow x 4.0) (pow y 4.0)))