Average Error: 14.7 → 10.8
Time: 21.8s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.97827715461108799631718450871196748365 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z} - \frac{1}{a - z} \cdot x, y - z, x\right)\\ \mathbf{elif}\;a \le 1.683145081775183698937570113592812053922 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - y \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}} + x\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.97827715461108799631718450871196748365 \cdot 10^{-73}:\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{a - z} - \frac{1}{a - z} \cdot x, y - z, x\right)\\

\mathbf{elif}\;a \le 1.683145081775183698937570113592812053922 \cdot 10^{-163}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - y \cdot \frac{t}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}} + x\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r81590 = x;
        double r81591 = y;
        double r81592 = z;
        double r81593 = r81591 - r81592;
        double r81594 = t;
        double r81595 = r81594 - r81590;
        double r81596 = a;
        double r81597 = r81596 - r81592;
        double r81598 = r81595 / r81597;
        double r81599 = r81593 * r81598;
        double r81600 = r81590 + r81599;
        return r81600;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r81601 = a;
        double r81602 = -1.978277154611088e-73;
        bool r81603 = r81601 <= r81602;
        double r81604 = t;
        double r81605 = z;
        double r81606 = r81601 - r81605;
        double r81607 = r81604 / r81606;
        double r81608 = 1.0;
        double r81609 = r81608 / r81606;
        double r81610 = x;
        double r81611 = r81609 * r81610;
        double r81612 = r81607 - r81611;
        double r81613 = y;
        double r81614 = r81613 - r81605;
        double r81615 = fma(r81612, r81614, r81610);
        double r81616 = 1.6831450817751837e-163;
        bool r81617 = r81601 <= r81616;
        double r81618 = r81610 / r81605;
        double r81619 = fma(r81618, r81613, r81604);
        double r81620 = r81604 / r81605;
        double r81621 = r81613 * r81620;
        double r81622 = r81619 - r81621;
        double r81623 = cbrt(r81606);
        double r81624 = r81623 * r81623;
        double r81625 = r81614 / r81624;
        double r81626 = r81604 - r81610;
        double r81627 = r81626 / r81623;
        double r81628 = r81625 * r81627;
        double r81629 = r81628 + r81610;
        double r81630 = r81617 ? r81622 : r81629;
        double r81631 = r81603 ? r81615 : r81630;
        return r81631;
}

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 < -1.978277154611088e-73

    1. Initial program 9.1

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

    2. Simplified9.0

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

    4. Applied div-sub9.0

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

    6. Applied div-inv9.0

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

    if -1.978277154611088e-73 < a < 1.6831450817751837e-163

    1. Initial program 24.6

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

    2. Taylor expanded around inf 15.8

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

    3. Simplified13.2

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

    if 1.6831450817751837e-163 < a

    1. Initial program 12.6

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

    3. Applied add-cube-cbrt13.1

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

    4. Applied *-un-lft-identity13.1

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

    5. Applied times-frac13.1

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

    6. Applied associate-*r*10.7

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

    7. Simplified10.7

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.97827715461108799631718450871196748365 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z} - \frac{1}{a - z} \cdot x, y - z, x\right)\\ \mathbf{elif}\;a \le 1.683145081775183698937570113592812053922 \cdot 10^{-163}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - y \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  (+ x (* (- y z) (/ (- t x) (- a z)))))