Average Error: 6.3 → 1.8
Time: 20.1s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(z - x\right) \cdot y}{t} \le -1.2244164738069483 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(z - x\right) \cdot y}{t} \le 1.9317375006271428 \cdot 10^{-122}:\\ \;\;\;\;\left(\frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{t}}} \cdot \left(\frac{z - x}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right)\right) \cdot \frac{1}{\sqrt[3]{t}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{\left(z - x\right) \cdot y}{t} \le -1.2244164738069483 \cdot 10^{-103}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\mathbf{elif}\;x + \frac{\left(z - x\right) \cdot y}{t} \le 1.9317375006271428 \cdot 10^{-122}:\\
\;\;\;\;\left(\frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{t}}} \cdot \left(\frac{z - x}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right)\right) \cdot \frac{1}{\sqrt[3]{t}} + x\\

\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 r13300364 = x;
        double r13300365 = y;
        double r13300366 = z;
        double r13300367 = r13300366 - r13300364;
        double r13300368 = r13300365 * r13300367;
        double r13300369 = t;
        double r13300370 = r13300368 / r13300369;
        double r13300371 = r13300364 + r13300370;
        return r13300371;
}
↓
double f(double x, double y, double z, double t) {
        double r13300372 = x;
        double r13300373 = z;
        double r13300374 = r13300373 - r13300372;
        double r13300375 = y;
        double r13300376 = r13300374 * r13300375;
        double r13300377 = t;
        double r13300378 = r13300376 / r13300377;
        double r13300379 = r13300372 + r13300378;
        double r13300380 = -1.2244164738069483e-103;
        bool r13300381 = r13300379 <= r13300380;
        double r13300382 = r13300375 / r13300377;
        double r13300383 = fma(r13300382, r13300374, r13300372);
        double r13300384 = 1.9317375006271428e-122;
        bool r13300385 = r13300379 <= r13300384;
        double r13300386 = cbrt(r13300375);
        double r13300387 = cbrt(r13300377);
        double r13300388 = cbrt(r13300387);
        double r13300389 = r13300386 / r13300388;
        double r13300390 = r13300374 / r13300387;
        double r13300391 = r13300386 * r13300386;
        double r13300392 = r13300387 * r13300387;
        double r13300393 = cbrt(r13300392);
        double r13300394 = r13300391 / r13300393;
        double r13300395 = r13300390 * r13300394;
        double r13300396 = r13300389 * r13300395;
        double r13300397 = 1.0;
        double r13300398 = r13300397 / r13300387;
        double r13300399 = r13300396 * r13300398;
        double r13300400 = r13300399 + r13300372;
        double r13300401 = r13300385 ? r13300400 : r13300383;
        double r13300402 = r13300381 ? r13300383 : r13300401;
        return r13300402;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.3
Target2.0
Herbie1.8
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (+ x (/ (* y (- z x)) t)) < -1.2244164738069483e-103 or 1.9317375006271428e-122 < (+ x (/ (* y (- z x)) t))

    1. Initial program 6.8

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

    2. Taylor expanded around 0 6.8

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

    3. Simplified1.8

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

    if -1.2244164738069483e-103 < (+ x (/ (* y (- z x)) t)) < 1.9317375006271428e-122

    1. Initial program 3.0

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

    3. Applied div-inv3.1

      \[\leadsto x + \color{blue}{\left(y \cdot \left(z - x\right)\right) \cdot \frac{1}{t}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt3.3

      \[\leadsto x + \left(y \cdot \left(z - x\right)\right) \cdot \frac{1}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}\]

    6. Applied *-un-lft-identity3.3

      \[\leadsto x + \left(y \cdot \left(z - x\right)\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}\]

    7. Applied times-frac3.3

      \[\leadsto x + \left(y \cdot \left(z - x\right)\right) \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{1}{\sqrt[3]{t}}\right)}\]

    8. Applied associate-*r*3.3

      \[\leadsto x + \color{blue}{\left(\left(y \cdot \left(z - x\right)\right) \cdot \frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{1}{\sqrt[3]{t}}}\]

    9. Simplified1.5

      \[\leadsto x + \color{blue}{\left(\frac{z - x}{\sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{t}}\right)} \cdot \frac{1}{\sqrt[3]{t}}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt1.6

      \[\leadsto x + \left(\frac{z - x}{\sqrt[3]{t}} \cdot \frac{y}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}}}\right) \cdot \frac{1}{\sqrt[3]{t}}\]

    12. Applied cbrt-prod1.6

      \[\leadsto x + \left(\frac{z - x}{\sqrt[3]{t}} \cdot \frac{y}{\color{blue}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}}\right) \cdot \frac{1}{\sqrt[3]{t}}\]

    13. Applied add-cube-cbrt1.7

      \[\leadsto x + \left(\frac{z - x}{\sqrt[3]{t}} \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \sqrt[3]{\sqrt[3]{t}}}\right) \cdot \frac{1}{\sqrt[3]{t}}\]

    14. Applied times-frac1.6

      \[\leadsto x + \left(\frac{z - x}{\sqrt[3]{t}} \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{t}}}\right)}\right) \cdot \frac{1}{\sqrt[3]{t}}\]

    15. Applied associate-*r*1.2

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(z - x\right) \cdot y}{t} \le -1.2244164738069483 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(z - x\right) \cdot y}{t} \le 1.9317375006271428 \cdot 10^{-122}:\\ \;\;\;\;\left(\frac{\sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{t}}} \cdot \left(\frac{z - x}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right)\right) \cdot \frac{1}{\sqrt[3]{t}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"

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

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