Average Error: 14.7 → 11.7
Time: 4.1s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.60032068403697767 \cdot 10^{103} \lor \neg \left(z \le 4.25828813818233938 \cdot 10^{216}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \left(t - x\right) \cdot \frac{1}{a - z}, x\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.60032068403697767 \cdot 10^{103} \lor \neg \left(z \le 4.25828813818233938 \cdot 10^{216}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r128413 = x;
        double r128414 = y;
        double r128415 = z;
        double r128416 = r128414 - r128415;
        double r128417 = t;
        double r128418 = r128417 - r128413;
        double r128419 = a;
        double r128420 = r128419 - r128415;
        double r128421 = r128418 / r128420;
        double r128422 = r128416 * r128421;
        double r128423 = r128413 + r128422;
        return r128423;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r128424 = z;
        double r128425 = -5.600320684036978e+103;
        bool r128426 = r128424 <= r128425;
        double r128427 = 4.2582881381823394e+216;
        bool r128428 = r128424 <= r128427;
        double r128429 = !r128428;
        bool r128430 = r128426 || r128429;
        double r128431 = y;
        double r128432 = x;
        double r128433 = r128432 / r128424;
        double r128434 = t;
        double r128435 = r128434 / r128424;
        double r128436 = r128433 - r128435;
        double r128437 = fma(r128431, r128436, r128434);
        double r128438 = r128431 - r128424;
        double r128439 = r128434 - r128432;
        double r128440 = 1.0;
        double r128441 = a;
        double r128442 = r128441 - r128424;
        double r128443 = r128440 / r128442;
        double r128444 = r128439 * r128443;
        double r128445 = fma(r128438, r128444, r128432);
        double r128446 = r128430 ? r128437 : r128445;
        return r128446;
}

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 < -5.600320684036978e+103 or 4.2582881381823394e+216 < z

    1. Initial program 27.2

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

    2. Simplified27.1

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

    3. Taylor expanded around inf 25.1

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

    4. Simplified16.3

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

    if -5.600320684036978e+103 < z < 4.2582881381823394e+216

    1. Initial program 10.0

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

    2. Simplified9.9

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

    4. Applied div-inv10.0

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

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.60032068403697767 \cdot 10^{103} \lor \neg \left(z \le 4.25828813818233938 \cdot 10^{216}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \left(t - x\right) \cdot \frac{1}{a - z}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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)))))