Average Error: 5.9 → 2.8
Time: 2.8s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[t \cdot \frac{y}{a} + \mathsf{fma}\left(-z, \frac{y}{a}, x\right)\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
t \cdot \frac{y}{a} + \mathsf{fma}\left(-z, \frac{y}{a}, x\right)
double f(double x, double y, double z, double t, double a) {
        double r440303 = x;
        double r440304 = y;
        double r440305 = z;
        double r440306 = t;
        double r440307 = r440305 - r440306;
        double r440308 = r440304 * r440307;
        double r440309 = a;
        double r440310 = r440308 / r440309;
        double r440311 = r440303 - r440310;
        return r440311;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r440312 = t;
        double r440313 = y;
        double r440314 = a;
        double r440315 = r440313 / r440314;
        double r440316 = r440312 * r440315;
        double r440317 = z;
        double r440318 = -r440317;
        double r440319 = x;
        double r440320 = fma(r440318, r440315, r440319);
        double r440321 = r440316 + r440320;
        return r440321;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original5.9
Target0.8
Herbie2.8
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Initial program 5.9

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

  2. Simplified2.8

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

  4. Applied fma-udef2.8

    \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - z\right) + x}\]
  5. Using strategy rm

  6. Applied sub-neg2.8

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

  7. Applied distribute-rgt-in2.8

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

  8. Applied associate-+l+2.8

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

  9. Simplified2.8

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

  10. Final simplification2.8

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

Reproduce

herbie shell --seed 2020057 +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)))