Average Error: 0.0 → 0.3
Time: 12.0s
Precision: 64
\[\left(x + y\right) \cdot \left(z + 1\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z + 1 \le -1.284674362912158862570777273504063487053:\\ \;\;\;\;\mathsf{fma}\left(x, z, y \cdot \left(z + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y\right) \cdot \sqrt{z + 1}\right) \cdot \sqrt{z + 1}\\ \end{array}\]
\left(x + y\right) \cdot \left(z + 1\right)
↓
\begin{array}{l}
\mathbf{if}\;z + 1 \le -1.284674362912158862570777273504063487053:\\
\;\;\;\;\mathsf{fma}\left(x, z, y \cdot \left(z + 1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\left(x + y\right) \cdot \sqrt{z + 1}\right) \cdot \sqrt{z + 1}\\

\end{array}
double f(double x, double y, double z) {
        double r45342 = x;
        double r45343 = y;
        double r45344 = r45342 + r45343;
        double r45345 = z;
        double r45346 = 1.0;
        double r45347 = r45345 + r45346;
        double r45348 = r45344 * r45347;
        return r45348;
}
↓
double f(double x, double y, double z) {
        double r45349 = z;
        double r45350 = 1.0;
        double r45351 = r45349 + r45350;
        double r45352 = -1.2846743629121589;
        bool r45353 = r45351 <= r45352;
        double r45354 = x;
        double r45355 = y;
        double r45356 = r45355 * r45351;
        double r45357 = fma(r45354, r45349, r45356);
        double r45358 = r45354 + r45355;
        double r45359 = sqrt(r45351);
        double r45360 = r45358 * r45359;
        double r45361 = r45360 * r45359;
        double r45362 = r45353 ? r45357 : r45361;
        return r45362;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if (+ z 1.0) < -1.2846743629121589

    1. Initial program 0.0

      \[\left(x + y\right) \cdot \left(z + 1\right)\]

    2. Taylor expanded around inf 1.0

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

    3. Simplified1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z, y \cdot \left(z + 1\right)\right)}\]

    if -1.2846743629121589 < (+ z 1.0)

    1. Initial program 0.0

      \[\left(x + y\right) \cdot \left(z + 1\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.2

      \[\leadsto \left(x + y\right) \cdot \color{blue}{\left(\sqrt{z + 1} \cdot \sqrt{z + 1}\right)}\]

    4. Applied associate-*r*0.2

      \[\leadsto \color{blue}{\left(\left(x + y\right) \cdot \sqrt{z + 1}\right) \cdot \sqrt{z + 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z + 1 \le -1.284674362912158862570777273504063487053:\\ \;\;\;\;\mathsf{fma}\left(x, z, y \cdot \left(z + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + y\right) \cdot \sqrt{z + 1}\right) \cdot \sqrt{z + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y z)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, G"
  :precision binary64
  (* (+ x y) (+ z 1)))