Average Error: 14.4 → 11.4
Time: 19.2s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.6084778564962155 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t}{z} \cdot y\\ \mathbf{elif}\;z \le 6.562213605057352 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t}{z} \cdot y\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.6084778564962155 \cdot 10^{+104}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t}{z} \cdot y\\

\mathbf{elif}\;z \le 6.562213605057352 \cdot 10^{+178}:\\
\;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r6400678 = x;
        double r6400679 = y;
        double r6400680 = z;
        double r6400681 = r6400679 - r6400680;
        double r6400682 = t;
        double r6400683 = r6400682 - r6400678;
        double r6400684 = a;
        double r6400685 = r6400684 - r6400680;
        double r6400686 = r6400683 / r6400685;
        double r6400687 = r6400681 * r6400686;
        double r6400688 = r6400678 + r6400687;
        return r6400688;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r6400689 = z;
        double r6400690 = -5.6084778564962155e+104;
        bool r6400691 = r6400689 <= r6400690;
        double r6400692 = x;
        double r6400693 = r6400692 / r6400689;
        double r6400694 = y;
        double r6400695 = t;
        double r6400696 = fma(r6400693, r6400694, r6400695);
        double r6400697 = r6400695 / r6400689;
        double r6400698 = r6400697 * r6400694;
        double r6400699 = r6400696 - r6400698;
        double r6400700 = 6.562213605057352e+178;
        bool r6400701 = r6400689 <= r6400700;
        double r6400702 = r6400694 - r6400689;
        double r6400703 = r6400695 - r6400692;
        double r6400704 = a;
        double r6400705 = r6400704 - r6400689;
        double r6400706 = r6400703 / r6400705;
        double r6400707 = fma(r6400702, r6400706, r6400692);
        double r6400708 = r6400701 ? r6400707 : r6400699;
        double r6400709 = r6400691 ? r6400699 : r6400708;
        return r6400709;
}

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 < -5.6084778564962155e+104 or 6.562213605057352e+178 < z

    1. Initial program 26.8

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

    2. Simplified26.7

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

    4. Applied fma-udef26.8

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

    5. Taylor expanded around inf 24.8

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

    6. Simplified17.0

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

    if -5.6084778564962155e+104 < z < 6.562213605057352e+178

    1. Initial program 9.1

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

    2. Simplified9.0

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

    4. Applied fma-udef9.1

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

    6. Applied fma-def9.0

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.6084778564962155 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t}{z} \cdot y\\ \mathbf{elif}\;z \le 6.562213605057352 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  (+ x (* (- y z) (/ (- t x) (- a z)))))