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

↓

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

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

↓

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

double f(double x, double y, double z, double t, double a) {
        double r311905 = x;
        double r311906 = y;
        double r311907 = z;
        double r311908 = t;
        double r311909 = r311907 - r311908;
        double r311910 = r311906 * r311909;
        double r311911 = a;
        double r311912 = r311910 / r311911;
        double r311913 = r311905 - r311912;
        return r311913;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r311914 = y;
        double r311915 = a;
        double r311916 = r311914 / r311915;
        double r311917 = t;
        double r311918 = z;
        double r311919 = r311917 - r311918;
        double r311920 = x;
        double r311921 = fma(r311916, r311919, r311920);
        return r311921;
}

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

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, F"
  :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)))