Average Error: 15.0 → 12.5
Time: 23.2s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -2.346627508952012515207561082871188905248 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(a - z\right) \cdot \frac{1}{t - x}}, y - z, x\right)\\ \mathbf{elif}\;a \le 7.030593888359200932512090940955049716269 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \le -2.346627508952012515207561082871188905248 \cdot 10^{-107}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\left(a - z\right) \cdot \frac{1}{t - x}}, y - z, x\right)\\

\mathbf{elif}\;a \le 7.030593888359200932512090940955049716269 \cdot 10^{-71}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r86962 = x;
        double r86963 = y;
        double r86964 = z;
        double r86965 = r86963 - r86964;
        double r86966 = t;
        double r86967 = r86966 - r86962;
        double r86968 = a;
        double r86969 = r86968 - r86964;
        double r86970 = r86967 / r86969;
        double r86971 = r86965 * r86970;
        double r86972 = r86962 + r86971;
        return r86972;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r86973 = a;
        double r86974 = -2.3466275089520125e-107;
        bool r86975 = r86973 <= r86974;
        double r86976 = 1.0;
        double r86977 = z;
        double r86978 = r86973 - r86977;
        double r86979 = t;
        double r86980 = x;
        double r86981 = r86979 - r86980;
        double r86982 = r86976 / r86981;
        double r86983 = r86978 * r86982;
        double r86984 = r86976 / r86983;
        double r86985 = y;
        double r86986 = r86985 - r86977;
        double r86987 = fma(r86984, r86986, r86980);
        double r86988 = 7.030593888359201e-71;
        bool r86989 = r86973 <= r86988;
        double r86990 = r86980 / r86977;
        double r86991 = r86979 * r86985;
        double r86992 = r86991 / r86977;
        double r86993 = r86979 - r86992;
        double r86994 = fma(r86990, r86985, r86993);
        double r86995 = r86976 / r86978;
        double r86996 = r86981 * r86995;
        double r86997 = r86986 * r86996;
        double r86998 = r86980 + r86997;
        double r86999 = r86989 ? r86994 : r86998;
        double r87000 = r86975 ? r86987 : r86999;
        return r87000;
}

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 < -2.3466275089520125e-107

    1. Initial program 11.8

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

    2. Simplified11.7

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

    4. Applied clear-num11.9

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

    6. Applied div-inv12.0

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

    if -2.3466275089520125e-107 < a < 7.030593888359201e-71

    1. Initial program 24.8

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

    2. Taylor expanded around inf 17.1

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

    3. Simplified16.6

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

    if 7.030593888359201e-71 < a

    1. Initial program 9.3

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

    3. Applied div-inv9.4

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

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.346627508952012515207561082871188905248 \cdot 10^{-107}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left(a - z\right) \cdot \frac{1}{t - x}}, y - z, x\right)\\ \mathbf{elif}\;a \le 7.030593888359200932512090940955049716269 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \left(\left(t - x\right) \cdot \frac{1}{a - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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)))))