Average Error: 6.7 → 2.0
Time: 30.5s
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 r413954 = x;
        double r413955 = y;
        double r413956 = z;
        double r413957 = r413956 - r413954;
        double r413958 = r413955 * r413957;
        double r413959 = t;
        double r413960 = r413958 / r413959;
        double r413961 = r413954 + r413960;
        return r413961;
}
↓
double f(double x, double y, double z, double t) {
        double r413962 = y;
        double r413963 = t;
        double r413964 = r413962 / r413963;
        double r413965 = z;
        double r413966 = x;
        double r413967 = r413965 - r413966;
        double r413968 = fma(r413964, r413967, r413966);
        return r413968;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.7
Target2.0
Herbie2.0
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Initial program 6.7

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

  2. Simplified2.0

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

  3. Final simplification2.0

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

Reproduce

herbie shell --seed 2020047 +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)))