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

↓

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

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

↓

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

double f(double x, double y, double z, double t, double a) {
        double r382484 = x;
        double r382485 = y;
        double r382486 = z;
        double r382487 = t;
        double r382488 = r382486 - r382487;
        double r382489 = r382485 * r382488;
        double r382490 = a;
        double r382491 = r382489 / r382490;
        double r382492 = r382484 + r382491;
        return r382492;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r382493 = y;
        double r382494 = a;
        double r382495 = r382493 / r382494;
        double r382496 = z;
        double r382497 = t;
        double r382498 = r382496 - r382497;
        double r382499 = x;
        double r382500 = fma(r382495, r382498, r382499);
        return r382500;
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Target

Original	6.2
Target	0.7
Herbie	2.5

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

Derivation

Initial program 6.2
\[x + \frac{y \cdot \left(z - t\right)}{a}\]
Simplified2.5
\[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{a}, z - t, x\right)}\]
Final simplification2.5
\[\leadsto \mathsf{fma}\left(\frac{y}{a}, z - t, x\right)\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

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