Average Error: 14.8 → 10.0
Time: 8.0s
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 r101975 = x;
        double r101976 = y;
        double r101977 = z;
        double r101978 = r101976 - r101977;
        double r101979 = t;
        double r101980 = r101979 - r101975;
        double r101981 = a;
        double r101982 = r101981 - r101977;
        double r101983 = r101980 / r101982;
        double r101984 = r101978 * r101983;
        double r101985 = r101975 + r101984;
        return r101985;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r101986 = z;
        double r101987 = -5.43006589941952e+202;
        bool r101988 = r101986 <= r101987;
        double r101989 = 4.1185661379444026e+185;
        bool r101990 = r101986 <= r101989;
        double r101991 = !r101990;
        bool r101992 = r101988 || r101991;
        double r101993 = y;
        double r101994 = x;
        double r101995 = r101994 / r101986;
        double r101996 = t;
        double r101997 = r101996 / r101986;
        double r101998 = r101995 - r101997;
        double r101999 = fma(r101993, r101998, r101996);
        double r102000 = 1.0;
        double r102001 = a;
        double r102002 = r102001 - r101986;
        double r102003 = r101996 - r101994;
        double r102004 = r102002 / r102003;
        double r102005 = cbrt(r102004);
        double r102006 = r102005 * r102005;
        double r102007 = r102000 / r102006;
        double r102008 = r101993 / r102005;
        double r102009 = r102007 * r102008;
        double r102010 = r101986 / r102004;
        double r102011 = r102010 - r101994;
        double r102012 = r102009 - r102011;
        double r102013 = r101992 ? r101999 : r102012;
        return r102013;
}

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)))))