Average Error: 14.8 → 14.0
Time: 24.8s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.153438144387475178721831385411661003915 \cdot 10^{194}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t - x\right) \cdot \frac{1}{a - z}, y - z, x\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.153438144387475178721831385411661003915 \cdot 10^{194}:\\
\;\;\;\;t\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r201629 = x;
        double r201630 = y;
        double r201631 = z;
        double r201632 = r201630 - r201631;
        double r201633 = t;
        double r201634 = r201633 - r201629;
        double r201635 = a;
        double r201636 = r201635 - r201631;
        double r201637 = r201634 / r201636;
        double r201638 = r201632 * r201637;
        double r201639 = r201629 + r201638;
        return r201639;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r201640 = z;
        double r201641 = -1.1534381443874752e+194;
        bool r201642 = r201640 <= r201641;
        double r201643 = t;
        double r201644 = x;
        double r201645 = r201643 - r201644;
        double r201646 = 1.0;
        double r201647 = a;
        double r201648 = r201647 - r201640;
        double r201649 = r201646 / r201648;
        double r201650 = r201645 * r201649;
        double r201651 = y;
        double r201652 = r201651 - r201640;
        double r201653 = fma(r201650, r201652, r201644);
        double r201654 = r201642 ? r201643 : r201653;
        return r201654;
}

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 2 regimes

  2. if z < -1.1534381443874752e+194

    1. Initial program 28.8

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

    2. Simplified28.6

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

    3. Taylor expanded around 0 21.3

      \[\leadsto \color{blue}{t}\]

    if -1.1534381443874752e+194 < z

    1. Initial program 13.2

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

    2. Simplified13.1

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

    4. Applied div-inv13.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(t - x\right) \cdot \frac{1}{a - z}}, y - z, x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification14.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.153438144387475178721831385411661003915 \cdot 10^{194}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(t - x\right) \cdot \frac{1}{a - z}, y - z, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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)))))