Average Error: 6.1 → 2.4
Time: 12.1s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\]
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 r199308 = x;
        double r199309 = y;
        double r199310 = z;
        double r199311 = t;
        double r199312 = r199310 - r199311;
        double r199313 = r199309 * r199312;
        double r199314 = a;
        double r199315 = r199313 / r199314;
        double r199316 = r199308 - r199315;
        return r199316;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r199317 = y;
        double r199318 = a;
        double r199319 = r199317 / r199318;
        double r199320 = t;
        double r199321 = z;
        double r199322 = r199320 - r199321;
        double r199323 = x;
        double r199324 = fma(r199319, r199322, r199323);
        return r199324;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.1
Target0.7
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753216593153715602325729 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.894426862792089097262541964056085749132 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  2. Simplified2.4

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

  3. Final simplification2.4

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

Reproduce

herbie shell --seed 2019235 +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)))