Average Error: 6.3 → 0.5
Time: 19.0s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 1.532969307568607748245310550315122934408 \cdot 10^{298}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - z}{a} \cdot y + x\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\

\mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 1.532969307568607748245310550315122934408 \cdot 10^{298}:\\
\;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{t - z}{a} \cdot y + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r280329 = x;
        double r280330 = y;
        double r280331 = z;
        double r280332 = t;
        double r280333 = r280331 - r280332;
        double r280334 = r280330 * r280333;
        double r280335 = a;
        double r280336 = r280334 / r280335;
        double r280337 = r280329 - r280336;
        return r280337;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r280338 = y;
        double r280339 = z;
        double r280340 = t;
        double r280341 = r280339 - r280340;
        double r280342 = r280338 * r280341;
        double r280343 = a;
        double r280344 = r280342 / r280343;
        double r280345 = -inf.0;
        bool r280346 = r280344 <= r280345;
        double r280347 = r280338 / r280343;
        double r280348 = r280340 - r280339;
        double r280349 = x;
        double r280350 = fma(r280347, r280348, r280349);
        double r280351 = 1.5329693075686077e+298;
        bool r280352 = r280344 <= r280351;
        double r280353 = r280349 - r280344;
        double r280354 = r280348 / r280343;
        double r280355 = r280354 * r280338;
        double r280356 = r280355 + r280349;
        double r280357 = r280352 ? r280353 : r280356;
        double r280358 = r280346 ? r280350 : r280357;
        return r280358;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.3
Target0.7
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (/ (* y (- z t)) a) < -inf.0

    1. Initial program 64.0

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

    2. Simplified0.2

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

    if -inf.0 < (/ (* y (- z t)) a) < 1.5329693075686077e+298

    1. Initial program 0.4

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

    if 1.5329693075686077e+298 < (/ (* y (- z t)) a)

    1. Initial program 58.5

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

    2. Simplified1.0

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

    4. Applied fma-udef1.0

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

    5. Simplified0.9

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

    7. Applied associate-/r/3.1

      \[\leadsto \color{blue}{\frac{t - z}{a} \cdot y} + x\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \le 1.532969307568607748245310550315122934408 \cdot 10^{298}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - z}{a} \cdot y + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :precision binary64

  :herbie-target
  (if (< y -1.07612662163899753e-10) (- x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.8944268627920891e-49) (- x (/ (* y (- z t)) a)) (- x (/ y (/ a (- z t))))))

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