Average Error: 6.8 → 1.5
Time: 21.9s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5805466880.89406108856201171875:\\ \;\;\;\;x + \frac{\frac{y}{t}}{\frac{1}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.760488666520461618655230049336452501407 \cdot 10^{300}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5805466880.89406108856201171875:\\
\;\;\;\;x + \frac{\frac{y}{t}}{\frac{1}{z - x}}\\

\mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.760488666520461618655230049336452501407 \cdot 10^{300}:\\
\;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r219287 = x;
        double r219288 = y;
        double r219289 = z;
        double r219290 = r219289 - r219287;
        double r219291 = r219288 * r219290;
        double r219292 = t;
        double r219293 = r219291 / r219292;
        double r219294 = r219287 + r219293;
        return r219294;
}
↓
double f(double x, double y, double z, double t) {
        double r219295 = x;
        double r219296 = y;
        double r219297 = z;
        double r219298 = r219297 - r219295;
        double r219299 = r219296 * r219298;
        double r219300 = t;
        double r219301 = r219299 / r219300;
        double r219302 = r219295 + r219301;
        double r219303 = 5805466880.894061;
        bool r219304 = r219302 <= r219303;
        double r219305 = r219296 / r219300;
        double r219306 = 1.0;
        double r219307 = r219306 / r219298;
        double r219308 = r219305 / r219307;
        double r219309 = r219295 + r219308;
        double r219310 = 5.760488666520462e+300;
        bool r219311 = r219302 <= r219310;
        double r219312 = fma(r219305, r219298, r219295);
        double r219313 = r219311 ? r219302 : r219312;
        double r219314 = r219304 ? r219309 : r219313;
        return r219314;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.8
Target2.1
Herbie1.5
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (+ x (/ (* y (- z x)) t)) < 5805466880.894061

    1. Initial program 5.8

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

    3. Applied associate-/l*4.9

      \[\leadsto x + \color{blue}{\frac{y}{\frac{t}{z - x}}}\]
    4. Using strategy rm

    5. Applied div-inv4.9

      \[\leadsto x + \frac{y}{\color{blue}{t \cdot \frac{1}{z - x}}}\]

    6. Applied associate-/r*2.1

      \[\leadsto x + \color{blue}{\frac{\frac{y}{t}}{\frac{1}{z - x}}}\]

    if 5805466880.894061 < (+ x (/ (* y (- z x)) t)) < 5.760488666520462e+300

    1. Initial program 0.1

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

    if 5.760488666520462e+300 < (+ x (/ (* y (- z x)) t))

    1. Initial program 56.9

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

    2. Simplified1.1

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5805466880.89406108856201171875:\\ \;\;\;\;x + \frac{\frac{y}{t}}{\frac{1}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 5.760488666520461618655230049336452501407 \cdot 10^{300}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))