Average Error: 14.5 → 10.6
Time: 14.6s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -7.952712787195166505836610605816383165417 \cdot 10^{-100} \lor \neg \left(a \le 8.392440744553116362698086020495240028125 \cdot 10^{-55}\right):\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t \cdot y}{z}\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \le -7.952712787195166505836610605816383165417 \cdot 10^{-100} \lor \neg \left(a \le 8.392440744553116362698086020495240028125 \cdot 10^{-55}\right):\\
\;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t \cdot y}{z}\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r80587 = x;
        double r80588 = y;
        double r80589 = z;
        double r80590 = r80588 - r80589;
        double r80591 = t;
        double r80592 = r80591 - r80587;
        double r80593 = a;
        double r80594 = r80593 - r80589;
        double r80595 = r80592 / r80594;
        double r80596 = r80590 * r80595;
        double r80597 = r80587 + r80596;
        return r80597;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r80598 = a;
        double r80599 = -7.952712787195167e-100;
        bool r80600 = r80598 <= r80599;
        double r80601 = 8.392440744553116e-55;
        bool r80602 = r80598 <= r80601;
        double r80603 = !r80602;
        bool r80604 = r80600 || r80603;
        double r80605 = x;
        double r80606 = y;
        double r80607 = z;
        double r80608 = r80606 - r80607;
        double r80609 = t;
        double r80610 = r80609 - r80605;
        double r80611 = cbrt(r80610);
        double r80612 = r80611 * r80611;
        double r80613 = r80598 - r80607;
        double r80614 = cbrt(r80613);
        double r80615 = r80614 * r80614;
        double r80616 = r80612 / r80615;
        double r80617 = r80608 * r80616;
        double r80618 = r80611 / r80614;
        double r80619 = r80617 * r80618;
        double r80620 = r80605 + r80619;
        double r80621 = r80605 / r80607;
        double r80622 = fma(r80621, r80606, r80609);
        double r80623 = r80609 * r80606;
        double r80624 = r80623 / r80607;
        double r80625 = r80622 - r80624;
        double r80626 = r80604 ? r80620 : r80625;
        return r80626;
}

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 a < -7.952712787195167e-100 or 8.392440744553116e-55 < a

    1. Initial program 9.8

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

    3. Applied add-cube-cbrt10.3

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

    4. Applied add-cube-cbrt10.4

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

    5. Applied times-frac10.5

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

    6. Applied associate-*r*8.3

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

    if -7.952712787195167e-100 < a < 8.392440744553116e-55

    1. Initial program 23.9

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

    2. Taylor expanded around inf 17.0

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

    3. Simplified15.3

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -7.952712787195166505836610605816383165417 \cdot 10^{-100} \lor \neg \left(a \le 8.392440744553116362698086020495240028125 \cdot 10^{-55}\right):\\ \;\;\;\;x + \left(\left(y - z\right) \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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)))))