Average Error: 6.6 → 2.0
Time: 17.7s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.099112012067726490279229104614636285691 \cdot 10^{-184} \lor \neg \left(x \le 1.277732222936873472254206748421226743137 \cdot 10^{-188}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y + y \cdot \left(-x\right)}{t}\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.099112012067726490279229104614636285691 \cdot 10^{-184} \lor \neg \left(x \le 1.277732222936873472254206748421226743137 \cdot 10^{-188}\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{z \cdot y + y \cdot \left(-x\right)}{t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r154234 = x;
        double r154235 = y;
        double r154236 = z;
        double r154237 = r154236 - r154234;
        double r154238 = r154235 * r154237;
        double r154239 = t;
        double r154240 = r154238 / r154239;
        double r154241 = r154234 + r154240;
        return r154241;
}
↓
double f(double x, double y, double z, double t) {
        double r154242 = x;
        double r154243 = -8.099112012067726e-184;
        bool r154244 = r154242 <= r154243;
        double r154245 = 1.2777322229368735e-188;
        bool r154246 = r154242 <= r154245;
        double r154247 = !r154246;
        bool r154248 = r154244 || r154247;
        double r154249 = y;
        double r154250 = t;
        double r154251 = r154249 / r154250;
        double r154252 = z;
        double r154253 = r154252 - r154242;
        double r154254 = fma(r154251, r154253, r154242);
        double r154255 = r154252 * r154249;
        double r154256 = -r154242;
        double r154257 = r154249 * r154256;
        double r154258 = r154255 + r154257;
        double r154259 = r154258 / r154250;
        double r154260 = r154242 + r154259;
        double r154261 = r154248 ? r154254 : r154260;
        return r154261;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.6
Target2.0
Herbie2.0
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -8.099112012067726e-184 or 1.2777322229368735e-188 < x

    1. Initial program 7.0

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]

    2. Simplified1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)}\]

    if -8.099112012067726e-184 < x < 1.2777322229368735e-188

    1. Initial program 5.4

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
    2. Using strategy rm

    3. Applied sub-neg5.4

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

    4. Applied distribute-lft-in5.4

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

    5. Simplified5.4

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

  4. Final simplification2.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.099112012067726490279229104614636285691 \cdot 10^{-184} \lor \neg \left(x \le 1.277732222936873472254206748421226743137 \cdot 10^{-188}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot y + y \cdot \left(-x\right)}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))