Average Error: 0.1 → 10.6
Time: 3.1s
Precision: binary64
\[\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -2.710148075684743 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y + y, y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + \left(e^{\mathsf{log1p}\left(\frac{y \cdot \left(y + y\right)}{y}\right)} - 1\right)\right)\right)\\ \end{array} \]
\left(\left(x \cdot x + y \cdot y\right) + y \cdot y\right) + y \cdot y
↓
\begin{array}{l}
\mathbf{if}\;x \leq -2.710148075684743 \cdot 10^{-281}:\\
\;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y + y, y \cdot y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + \left(e^{\mathsf{log1p}\left(\frac{y \cdot \left(y + y\right)}{y}\right)} - 1\right)\right)\right)\\


\end{array}

(FPCore (x y) :precision binary64 (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))
↓
(FPCore (x y)
 :precision binary64
 (if (<= x -2.710148075684743e-281)
   (fma x x (fma y (+ y y) (* y y)))
   (fma x x (* y (+ y (- (exp (log1p (/ (* y (+ y y)) y))) 1.0))))))
double code(double x, double y) {
	return (((x * x) + (y * y)) + (y * y)) + (y * y);
}
↓
double code(double x, double y) {
	double tmp;
	if (x <= -2.710148075684743e-281) {
		tmp = fma(x, x, fma(y, (y + y), (y * y)));
	} else {
		tmp = fma(x, x, (y * (y + (exp(log1p(((y * (y + y)) / y))) - 1.0))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.1
Target0.1
Herbie10.6
\[x \cdot x + y \cdot \left(y + \left(y + y\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.7101480756847431e-281

    1. Initial program 0.1

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

    2. Simplified0.1

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

    if -2.7101480756847431e-281 < x

    1. Initial program 0.1

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

    2. Simplified0.1

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

    3. Applied egg-rr0.1

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

    4. Applied egg-rr19.9

      \[\leadsto \mathsf{fma}\left(x, x, y \cdot \left(y + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y \cdot \left(y + y\right)}{y}\right)} - 1\right)}\right)\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.710148075684743 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(x, x, \mathsf{fma}\left(y, y + y, y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, x, y \cdot \left(y + \left(e^{\mathsf{log1p}\left(\frac{y \cdot \left(y + y\right)}{y}\right)} - 1\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022127 
(FPCore (x y)
  :name "Linear.Quaternion:$c/ from linear-1.19.1.3, E"
  :precision binary64

  :herbie-target
  (+ (* x x) (* y (+ y (+ y y))))

  (+ (+ (+ (* x x) (* y y)) (* y y)) (* y y)))