Average Error: 6.0 → 1.4
Time: 9.0s
Precision: 64
\[x - \frac{y \cdot \left(z - t\right)}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.8751699245657391 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\ \mathbf{elif}\;a \le 6.102226148835496 \cdot 10^{-161}:\\ \;\;\;\;\left(x + \frac{t \cdot y}{a}\right) - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]
x - \frac{y \cdot \left(z - t\right)}{a}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.8751699245657391 \cdot 10^{-66}:\\
\;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\

\mathbf{elif}\;a \le 6.102226148835496 \cdot 10^{-161}:\\
\;\;\;\;\left(x + \frac{t \cdot y}{a}\right) - \frac{z \cdot y}{a}\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r423256 = x;
        double r423257 = y;
        double r423258 = z;
        double r423259 = t;
        double r423260 = r423258 - r423259;
        double r423261 = r423257 * r423260;
        double r423262 = a;
        double r423263 = r423261 / r423262;
        double r423264 = r423256 - r423263;
        return r423264;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r423265 = a;
        double r423266 = -1.875169924565739e-66;
        bool r423267 = r423265 <= r423266;
        double r423268 = y;
        double r423269 = t;
        double r423270 = z;
        double r423271 = r423269 - r423270;
        double r423272 = r423265 / r423271;
        double r423273 = r423268 / r423272;
        double r423274 = x;
        double r423275 = r423273 + r423274;
        double r423276 = 6.102226148835496e-161;
        bool r423277 = r423265 <= r423276;
        double r423278 = r423269 * r423268;
        double r423279 = r423278 / r423265;
        double r423280 = r423274 + r423279;
        double r423281 = r423270 * r423268;
        double r423282 = r423281 / r423265;
        double r423283 = r423280 - r423282;
        double r423284 = r423268 / r423265;
        double r423285 = fma(r423284, r423271, r423274);
        double r423286 = r423277 ? r423283 : r423285;
        double r423287 = r423267 ? r423275 : r423286;
        return r423287;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original6.0
Target0.6
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;y \lt -1.07612662163899753 \cdot 10^{-10}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y \lt 2.8944268627920891 \cdot 10^{-49}:\\ \;\;\;\;x - \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{a}{z - t}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if a < -1.875169924565739e-66

    1. Initial program 7.6

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

    2. Simplified0.9

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

    4. Applied clear-num1.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a}{t - z}}}, y, x\right)\]
    5. Using strategy rm

    6. Applied fma-udef1.0

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

    7. Simplified0.8

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

    if -1.875169924565739e-66 < a < 6.102226148835496e-161

    1. Initial program 1.2

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

    2. Simplified21.7

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

    4. Applied clear-num21.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a}{t - z}}}, y, x\right)\]
    5. Using strategy rm

    6. Applied fma-udef21.8

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

    7. Simplified20.0

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

    8. Taylor expanded around 0 1.2

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

    if 6.102226148835496e-161 < a

    1. Initial program 6.8

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

    2. Simplified3.0

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

    4. Applied clear-num3.1

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{a}{t - z}}}, y, x\right)\]
    5. Using strategy rm

    6. Applied fma-udef3.1

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

    7. Simplified2.7

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

    9. Applied associate-/r/2.0

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

    10. Applied fma-def2.0

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.8751699245657391 \cdot 10^{-66}:\\ \;\;\;\;\frac{y}{\frac{a}{t - z}} + x\\ \mathbf{elif}\;a \le 6.102226148835496 \cdot 10^{-161}:\\ \;\;\;\;\left(x + \frac{t \cdot y}{a}\right) - \frac{z \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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)))