Average Error: 5.9 → 0.9
Time: 2.7s
Precision: 64
\[x + \frac{y \cdot \left(z - x\right)}{t}\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + \frac{\frac{y}{t}}{\frac{1}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.6537122954451224 \cdot 10^{266}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(z - x\right)}{t}
↓
\begin{array}{l}
\mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\
\;\;\;\;x + \frac{\frac{y}{t}}{\frac{1}{z - x}}\\

\mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.6537122954451224 \cdot 10^{266}:\\
\;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r303238 = x;
        double r303239 = y;
        double r303240 = z;
        double r303241 = r303240 - r303238;
        double r303242 = r303239 * r303241;
        double r303243 = t;
        double r303244 = r303242 / r303243;
        double r303245 = r303238 + r303244;
        return r303245;
}
↓
double f(double x, double y, double z, double t) {
        double r303246 = x;
        double r303247 = y;
        double r303248 = z;
        double r303249 = r303248 - r303246;
        double r303250 = r303247 * r303249;
        double r303251 = t;
        double r303252 = r303250 / r303251;
        double r303253 = r303246 + r303252;
        double r303254 = -inf.0;
        bool r303255 = r303253 <= r303254;
        double r303256 = r303247 / r303251;
        double r303257 = 1.0;
        double r303258 = r303257 / r303249;
        double r303259 = r303256 / r303258;
        double r303260 = r303246 + r303259;
        double r303261 = 8.653712295445122e+266;
        bool r303262 = r303253 <= r303261;
        double r303263 = fma(r303256, r303249, r303246);
        double r303264 = r303262 ? r303253 : r303263;
        double r303265 = r303255 ? r303260 : r303264;
        return r303265;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original5.9
Target2.0
Herbie0.9
\[x - \left(x \cdot \frac{y}{t} + \left(-z\right) \cdot \frac{y}{t}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (+ x (/ (* y (- z x)) t)) < -inf.0

    1. Initial program 64.0

      \[x + \frac{y \cdot \left(z - x\right)}{t}\]
    2. Using strategy rm

    3. Applied associate-/l*0.2

      \[\leadsto x + \color{blue}{\frac{y}{\frac{t}{z - x}}}\]
    4. Using strategy rm

    5. Applied div-inv0.3

      \[\leadsto x + \frac{y}{\color{blue}{t \cdot \frac{1}{z - x}}}\]

    6. Applied associate-/r*0.2

      \[\leadsto x + \color{blue}{\frac{\frac{y}{t}}{\frac{1}{z - x}}}\]

    if -inf.0 < (+ x (/ (* y (- z x)) t)) < 8.653712295445122e+266

    1. Initial program 0.7

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

    if 8.653712295445122e+266 < (+ x (/ (* y (- z x)) t))

    1. Initial program 31.9

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

    2. Simplified2.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} = -\infty:\\ \;\;\;\;x + \frac{\frac{y}{t}}{\frac{1}{z - x}}\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \le 8.6537122954451224 \cdot 10^{266}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +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)))