Average Error: 6.5 → 2.2
Time: 2.8s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)
double f(double x, double y, double z, double t) {
        double r303663 = x;
        double r303664 = y;
        double r303665 = z;
        double r303666 = r303665 - r303663;
        double r303667 = r303664 * r303666;
        double r303668 = t;
        double r303669 = r303667 / r303668;
        double r303670 = r303663 + r303669;
        return r303670;
}
↓
double f(double x, double y, double z, double t) {
        double r303671 = y;
        double r303672 = t;
        double r303673 = r303671 / r303672;
        double r303674 = z;
        double r303675 = x;
        double r303676 = r303674 - r303675;
        double r303677 = fma(r303673, r303676, r303675);
        return r303677;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.5
Target2.2
Herbie2.2
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Initial program 6.5

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

  2. Simplified2.2

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

  3. Final simplification2.2

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

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

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