Average Error: 14.4 → 11.4
Time: 17.8s
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 r6325915 = x;
        double r6325916 = y;
        double r6325917 = z;
        double r6325918 = r6325916 - r6325917;
        double r6325919 = t;
        double r6325920 = r6325919 - r6325915;
        double r6325921 = a;
        double r6325922 = r6325921 - r6325917;
        double r6325923 = r6325920 / r6325922;
        double r6325924 = r6325918 * r6325923;
        double r6325925 = r6325915 + r6325924;
        return r6325925;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r6325926 = z;
        double r6325927 = -5.6084778564962155e+104;
        bool r6325928 = r6325926 <= r6325927;
        double r6325929 = x;
        double r6325930 = r6325929 / r6325926;
        double r6325931 = y;
        double r6325932 = t;
        double r6325933 = fma(r6325930, r6325931, r6325932);
        double r6325934 = r6325932 / r6325926;
        double r6325935 = r6325934 * r6325931;
        double r6325936 = r6325933 - r6325935;
        double r6325937 = 6.562213605057352e+178;
        bool r6325938 = r6325926 <= r6325937;
        double r6325939 = r6325931 - r6325926;
        double r6325940 = r6325932 - r6325929;
        double r6325941 = a;
        double r6325942 = r6325941 - r6325926;
        double r6325943 = r6325940 / r6325942;
        double r6325944 = fma(r6325939, r6325943, r6325929);
        double r6325945 = r6325938 ? r6325944 : r6325936;
        double r6325946 = r6325928 ? r6325936 : r6325945;
        return r6325946;
}

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)))))