Average Error: 14.8 → 10.2
Time: 5.5s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -8.8291030278481248 \cdot 10^{84} \lor \neg \left(z \le 2.558230882618436 \cdot 10^{163}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a - z}{t - x}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -8.8291030278481248 \cdot 10^{84} \lor \neg \left(z \le 2.558230882618436 \cdot 10^{163}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r137790 = x;
        double r137791 = y;
        double r137792 = z;
        double r137793 = r137791 - r137792;
        double r137794 = t;
        double r137795 = r137794 - r137790;
        double r137796 = a;
        double r137797 = r137796 - r137792;
        double r137798 = r137795 / r137797;
        double r137799 = r137793 * r137798;
        double r137800 = r137790 + r137799;
        return r137800;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r137801 = z;
        double r137802 = -8.829103027848125e+84;
        bool r137803 = r137801 <= r137802;
        double r137804 = 2.5582308826184364e+163;
        bool r137805 = r137801 <= r137804;
        double r137806 = !r137805;
        bool r137807 = r137803 || r137806;
        double r137808 = y;
        double r137809 = x;
        double r137810 = r137809 / r137801;
        double r137811 = t;
        double r137812 = r137811 / r137801;
        double r137813 = r137810 - r137812;
        double r137814 = fma(r137808, r137813, r137811);
        double r137815 = a;
        double r137816 = r137815 - r137801;
        double r137817 = r137811 - r137809;
        double r137818 = r137816 / r137817;
        double r137819 = r137808 / r137818;
        double r137820 = r137801 / r137818;
        double r137821 = r137820 - r137809;
        double r137822 = r137819 - r137821;
        double r137823 = r137807 ? r137814 : r137822;
        return r137823;
}

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 < -8.829103027848125e+84 or 2.5582308826184364e+163 < z

    1. Initial program 26.6

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

    2. Simplified26.4

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

    4. Applied clear-num27.0

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

    6. Applied fma-udef27.1

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

    7. Simplified27.0

      \[\leadsto \color{blue}{\frac{y - z}{\frac{a - z}{t - x}}} + x\]
    8. Using strategy rm

    9. Applied associate-/r/22.0

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

    10. Applied fma-def22.0

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

    11. Taylor expanded around inf 24.9

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

    12. Simplified16.5

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

    if -8.829103027848125e+84 < z < 2.5582308826184364e+163

    1. Initial program 9.0

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

    2. Simplified9.0

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

    4. Applied clear-num9.1

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

    6. Applied fma-udef9.2

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

    7. Simplified8.8

      \[\leadsto \color{blue}{\frac{y - z}{\frac{a - z}{t - x}}} + x\]
    8. Using strategy rm

    9. Applied div-sub8.8

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

    10. Applied associate-+l-7.1

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.8291030278481248 \cdot 10^{84} \lor \neg \left(z \le 2.558230882618436 \cdot 10^{163}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{a - z}{t - x}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 +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)))))