Average Error: 6.3 → 0.3
Time: 12.1s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -1.056329197442560351249015028822880627264 \cdot 10^{281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, \frac{-z}{\frac{a}{y}}\right) + x\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 1.301385111956770635951737168355727076246 \cdot 10^{272}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\left(z - t\right) \cdot y \le -1.056329197442560351249015028822880627264 \cdot 10^{281}:\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, \frac{-z}{\frac{a}{y}}\right) + x\\

\mathbf{elif}\;\left(z - t\right) \cdot y \le 1.301385111956770635951737168355727076246 \cdot 10^{272}:\\
\;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r228577 = x;
        double r228578 = y;
        double r228579 = z;
        double r228580 = t;
        double r228581 = r228579 - r228580;
        double r228582 = r228578 * r228581;
        double r228583 = a;
        double r228584 = r228582 / r228583;
        double r228585 = r228577 - r228584;
        return r228585;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r228586 = z;
        double r228587 = t;
        double r228588 = r228586 - r228587;
        double r228589 = y;
        double r228590 = r228588 * r228589;
        double r228591 = -1.0563291974425604e+281;
        bool r228592 = r228590 <= r228591;
        double r228593 = a;
        double r228594 = r228587 / r228593;
        double r228595 = -r228586;
        double r228596 = r228593 / r228589;
        double r228597 = r228595 / r228596;
        double r228598 = fma(r228594, r228589, r228597);
        double r228599 = x;
        double r228600 = r228598 + r228599;
        double r228601 = 1.3013851119567706e+272;
        bool r228602 = r228590 <= r228601;
        double r228603 = r228590 / r228593;
        double r228604 = r228599 - r228603;
        double r228605 = r228589 / r228593;
        double r228606 = r228587 - r228586;
        double r228607 = fma(r228605, r228606, r228599);
        double r228608 = r228602 ? r228604 : r228607;
        double r228609 = r228592 ? r228600 : r228608;
        return r228609;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.3
Target0.6
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* y (- z t)) < -1.0563291974425604e+281

    1. Initial program 49.5

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

    2. Simplified0.4

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

    4. Applied fma-udef0.4

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

    5. Simplified0.2

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

    6. Taylor expanded around 0 49.5

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

    7. Simplified0.2

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

    if -1.0563291974425604e+281 < (* y (- z t)) < 1.3013851119567706e+272

    1. Initial program 0.3

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

    if 1.3013851119567706e+272 < (* y (- z t))

    1. Initial program 48.9

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

    2. Simplified0.2

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z - t\right) \cdot y \le -1.056329197442560351249015028822880627264 \cdot 10^{281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, \frac{-z}{\frac{a}{y}}\right) + x\\ \mathbf{elif}\;\left(z - t\right) \cdot y \le 1.301385111956770635951737168355727076246 \cdot 10^{272}:\\ \;\;\;\;x - \frac{\left(z - t\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"

  :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)))