Average Error: 14.5 → 12.3
Time: 21.3s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -9.16612252681064813 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{t - x} - \frac{z}{t - x}}, y - z, x\right)\\ \mathbf{elif}\;a \le 2.8169560441423619 \cdot 10^{-147}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{a - z}}{\frac{1}{t - x}}, y - z, x\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \le -9.16612252681064813 \cdot 10^{-164}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{t - x} - \frac{z}{t - x}}, y - z, x\right)\\

\mathbf{elif}\;a \le 2.8169560441423619 \cdot 10^{-147}:\\
\;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{a - z}}{\frac{1}{t - x}}, y - z, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r115274 = x;
        double r115275 = y;
        double r115276 = z;
        double r115277 = r115275 - r115276;
        double r115278 = t;
        double r115279 = r115278 - r115274;
        double r115280 = a;
        double r115281 = r115280 - r115276;
        double r115282 = r115279 / r115281;
        double r115283 = r115277 * r115282;
        double r115284 = r115274 + r115283;
        return r115284;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r115285 = a;
        double r115286 = -9.166122526810648e-164;
        bool r115287 = r115285 <= r115286;
        double r115288 = 1.0;
        double r115289 = t;
        double r115290 = x;
        double r115291 = r115289 - r115290;
        double r115292 = r115285 / r115291;
        double r115293 = z;
        double r115294 = r115293 / r115291;
        double r115295 = r115292 - r115294;
        double r115296 = r115288 / r115295;
        double r115297 = y;
        double r115298 = r115297 - r115293;
        double r115299 = fma(r115296, r115298, r115290);
        double r115300 = 2.816956044142362e-147;
        bool r115301 = r115285 <= r115300;
        double r115302 = r115290 * r115297;
        double r115303 = r115302 / r115293;
        double r115304 = r115303 + r115289;
        double r115305 = r115289 * r115297;
        double r115306 = r115305 / r115293;
        double r115307 = r115304 - r115306;
        double r115308 = r115285 - r115293;
        double r115309 = r115288 / r115308;
        double r115310 = r115288 / r115291;
        double r115311 = r115309 / r115310;
        double r115312 = fma(r115311, r115298, r115290);
        double r115313 = r115301 ? r115307 : r115312;
        double r115314 = r115287 ? r115299 : r115313;
        return r115314;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Split input into 3 regimes

  2. if a < -9.166122526810648e-164

    1. Initial program 12.3

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

    2. Simplified12.3

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

    4. Applied clear-num12.5

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

    6. Applied div-sub12.7

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

    if -9.166122526810648e-164 < a < 2.816956044142362e-147

    1. Initial program 24.5

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

    2. Taylor expanded around inf 13.1

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

    if 2.816956044142362e-147 < a

    1. Initial program 11.2

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

    2. Simplified11.2

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

    4. Applied clear-num11.5

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

    6. Applied div-inv11.5

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\left(a - z\right) \cdot \frac{1}{t - x}}}, y - z, x\right)\]

    7. Applied associate-/r*11.3

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{a - z}}{\frac{1}{t - x}}}, y - z, x\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -9.16612252681064813 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{a}{t - x} - \frac{z}{t - x}}, y - z, x\right)\\ \mathbf{elif}\;a \le 2.8169560441423619 \cdot 10^{-147}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{1}{a - z}}{\frac{1}{t - x}}, y - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))