Average Error: 14.8 → 10.0
Time: 8.1s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.430065899419519609027369218758157205239 \cdot 10^{202} \lor \neg \left(z \le 4.118566137944402609284645272819524853321 \cdot 10^{185}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a - z}{t - x}} \cdot \sqrt[3]{\frac{a - z}{t - x}}} \cdot \frac{y}{\sqrt[3]{\frac{a - z}{t - x}}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.430065899419519609027369218758157205239 \cdot 10^{202} \lor \neg \left(z \le 4.118566137944402609284645272819524853321 \cdot 10^{185}\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt[3]{\frac{a - z}{t - x}} \cdot \sqrt[3]{\frac{a - z}{t - x}}} \cdot \frac{y}{\sqrt[3]{\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 r75623 = x;
        double r75624 = y;
        double r75625 = z;
        double r75626 = r75624 - r75625;
        double r75627 = t;
        double r75628 = r75627 - r75623;
        double r75629 = a;
        double r75630 = r75629 - r75625;
        double r75631 = r75628 / r75630;
        double r75632 = r75626 * r75631;
        double r75633 = r75623 + r75632;
        return r75633;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r75634 = z;
        double r75635 = -5.43006589941952e+202;
        bool r75636 = r75634 <= r75635;
        double r75637 = 4.1185661379444026e+185;
        bool r75638 = r75634 <= r75637;
        double r75639 = !r75638;
        bool r75640 = r75636 || r75639;
        double r75641 = y;
        double r75642 = x;
        double r75643 = r75642 / r75634;
        double r75644 = t;
        double r75645 = r75644 / r75634;
        double r75646 = r75643 - r75645;
        double r75647 = fma(r75641, r75646, r75644);
        double r75648 = 1.0;
        double r75649 = a;
        double r75650 = r75649 - r75634;
        double r75651 = r75644 - r75642;
        double r75652 = r75650 / r75651;
        double r75653 = cbrt(r75652);
        double r75654 = r75653 * r75653;
        double r75655 = r75648 / r75654;
        double r75656 = r75641 / r75653;
        double r75657 = r75655 * r75656;
        double r75658 = r75634 / r75652;
        double r75659 = r75658 - r75642;
        double r75660 = r75657 - r75659;
        double r75661 = r75640 ? r75647 : r75660;
        return r75661;
}

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 < -5.43006589941952e+202 or 4.1185661379444026e+185 < z

    1. Initial program 28.9

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

    2. Simplified28.8

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

    3. Taylor expanded around inf 24.0

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

    4. Simplified13.6

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

    if -5.43006589941952e+202 < z < 4.1185661379444026e+185

    1. Initial program 10.7

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

    2. Simplified10.7

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

    4. Applied clear-num10.9

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

    6. Applied fma-udef10.9

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

    7. Simplified10.6

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

    9. Applied div-sub10.6

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

    10. Applied associate-+l-8.6

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

    12. Applied add-cube-cbrt9.0

      \[\leadsto \frac{y}{\color{blue}{\left(\sqrt[3]{\frac{a - z}{t - x}} \cdot \sqrt[3]{\frac{a - z}{t - x}}\right) \cdot \sqrt[3]{\frac{a - z}{t - x}}}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\]

    13. Applied *-un-lft-identity9.0

      \[\leadsto \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{\frac{a - z}{t - x}} \cdot \sqrt[3]{\frac{a - z}{t - x}}\right) \cdot \sqrt[3]{\frac{a - z}{t - x}}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\]

    14. Applied times-frac9.0

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.430065899419519609027369218758157205239 \cdot 10^{202} \lor \neg \left(z \le 4.118566137944402609284645272819524853321 \cdot 10^{185}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z} - \frac{t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{a - z}{t - x}} \cdot \sqrt[3]{\frac{a - z}{t - x}}} \cdot \frac{y}{\sqrt[3]{\frac{a - z}{t - x}}} - \left(\frac{z}{\frac{a - z}{t - x}} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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)))))