Average Error: 5.7 → 0.6
Time: 20.3s
Precision: 64
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -6.4435085564630181 \cdot 10^{176}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, -\mathsf{fma}\left(\frac{t}{a}, y, -x\right)\right) - \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 1.59757199093709241 \cdot 10^{153}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a}{y}} - \left(\frac{t}{\frac{a}{y}} - x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot \left(z - t\right) \le -6.4435085564630181 \cdot 10^{176}:\\
\;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, -\mathsf{fma}\left(\frac{t}{a}, y, -x\right)\right) - \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\\

\mathbf{elif}\;y \cdot \left(z - t\right) \le 1.59757199093709241 \cdot 10^{153}:\\
\;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r226331 = x;
        double r226332 = y;
        double r226333 = z;
        double r226334 = t;
        double r226335 = r226333 - r226334;
        double r226336 = r226332 * r226335;
        double r226337 = a;
        double r226338 = r226336 / r226337;
        double r226339 = r226331 + r226338;
        return r226339;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r226340 = y;
        double r226341 = z;
        double r226342 = t;
        double r226343 = r226341 - r226342;
        double r226344 = r226340 * r226343;
        double r226345 = -6.443508556463018e+176;
        bool r226346 = r226344 <= r226345;
        double r226347 = a;
        double r226348 = r226341 / r226347;
        double r226349 = r226342 / r226347;
        double r226350 = x;
        double r226351 = -r226350;
        double r226352 = fma(r226349, r226340, r226351);
        double r226353 = -r226352;
        double r226354 = fma(r226348, r226340, r226353);
        double r226355 = cbrt(r226350);
        double r226356 = -r226355;
        double r226357 = r226355 * r226355;
        double r226358 = r226355 * r226357;
        double r226359 = fma(r226356, r226357, r226358);
        double r226360 = r226354 - r226359;
        double r226361 = 1.5975719909370924e+153;
        bool r226362 = r226344 <= r226361;
        double r226363 = r226344 / r226347;
        double r226364 = r226350 + r226363;
        double r226365 = r226347 / r226340;
        double r226366 = r226341 / r226365;
        double r226367 = r226342 / r226365;
        double r226368 = r226367 - r226350;
        double r226369 = r226366 - r226368;
        double r226370 = r226362 ? r226364 : r226369;
        double r226371 = r226346 ? r226360 : r226370;
        return r226371;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.7
Target0.6
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -6.443508556463018e+176

    1. Initial program 22.7

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

    2. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef0.9

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

    5. Simplified1.0

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

    7. Applied div-sub1.0

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

    8. Applied associate-+l-1.0

      \[\leadsto \color{blue}{\frac{z}{\frac{a}{y}} - \left(\frac{t}{\frac{a}{y}} - x\right)}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt1.3

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

    11. Applied associate-/r/1.8

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

    12. Applied prod-diff1.8

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

    13. Applied associate--r+1.8

      \[\leadsto \color{blue}{\left(\frac{z}{\frac{a}{y}} - \mathsf{fma}\left(\frac{t}{a}, y, -\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right) - \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}\]

    14. Simplified1.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{a}, y, -\mathsf{fma}\left(\frac{t}{a}, y, -x\right)\right)} - \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\]

    if -6.443508556463018e+176 < (* y (- z t)) < 1.5975719909370924e+153

    1. Initial program 0.4

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

    if 1.5975719909370924e+153 < (* y (- z t))

    1. Initial program 21.2

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

    2. Simplified0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef0.8

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

    5. Simplified0.8

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

    7. Applied div-sub0.8

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

    8. Applied associate-+l-0.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \le -6.4435085564630181 \cdot 10^{176}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y, -\mathsf{fma}\left(\frac{t}{a}, y, -x\right)\right) - \mathsf{fma}\left(-\sqrt[3]{x}, \sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \le 1.59757199093709241 \cdot 10^{153}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a}{y}} - \left(\frac{t}{\frac{a}{y}} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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