Average Error: 14.8 → 10.9
Time: 20.6s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -1.601642543210548858492050646487388138085 \cdot 10^{-114} \lor \neg \left(a \le 1.690098775090531459046383339387065140428 \cdot 10^{-104}\right):\\ \;\;\;\;x + \left(\left(\sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(y - z\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(y - z\right)}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(y - z\right)}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t \cdot y}{z}\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \le -1.601642543210548858492050646487388138085 \cdot 10^{-114} \lor \neg \left(a \le 1.690098775090531459046383339387065140428 \cdot 10^{-104}\right):\\
\;\;\;\;x + \left(\left(\sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(y - z\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(y - z\right)}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(y - z\right)}\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 r86843 = x;
        double r86844 = y;
        double r86845 = z;
        double r86846 = r86844 - r86845;
        double r86847 = t;
        double r86848 = r86847 - r86843;
        double r86849 = a;
        double r86850 = r86849 - r86845;
        double r86851 = r86848 / r86850;
        double r86852 = r86846 * r86851;
        double r86853 = r86843 + r86852;
        return r86853;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r86854 = a;
        double r86855 = -1.6016425432105489e-114;
        bool r86856 = r86854 <= r86855;
        double r86857 = 1.6900987750905315e-104;
        bool r86858 = r86854 <= r86857;
        double r86859 = !r86858;
        bool r86860 = r86856 || r86859;
        double r86861 = x;
        double r86862 = t;
        double r86863 = r86862 - r86861;
        double r86864 = cbrt(r86863);
        double r86865 = r86864 * r86864;
        double r86866 = z;
        double r86867 = r86854 - r86866;
        double r86868 = cbrt(r86867);
        double r86869 = r86868 * r86868;
        double r86870 = r86865 / r86869;
        double r86871 = y;
        double r86872 = r86871 - r86866;
        double r86873 = r86870 * r86872;
        double r86874 = cbrt(r86873);
        double r86875 = r86874 * r86874;
        double r86876 = r86875 * r86874;
        double r86877 = r86864 / r86868;
        double r86878 = r86876 * r86877;
        double r86879 = r86861 + r86878;
        double r86880 = r86861 / r86866;
        double r86881 = fma(r86880, r86871, r86862);
        double r86882 = r86862 * r86871;
        double r86883 = r86882 / r86866;
        double r86884 = r86881 - r86883;
        double r86885 = r86860 ? r86879 : r86884;
        return r86885;
}

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 < -1.6016425432105489e-114 or 1.6900987750905315e-104 < a

    1. Initial program 10.8

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

    3. Applied add-cube-cbrt11.2

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

    4. Applied add-cube-cbrt11.4

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

    5. Applied times-frac11.4

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

    6. Applied associate-*r*9.2

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

    7. Simplified9.2

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

    9. Applied add-cube-cbrt9.3

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

    if -1.6016425432105489e-114 < a < 1.6900987750905315e-104

    1. Initial program 24.5

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

    2. Taylor expanded around inf 15.9

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

    3. Simplified14.7

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

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.601642543210548858492050646487388138085 \cdot 10^{-114} \lor \neg \left(a \le 1.690098775090531459046383339387065140428 \cdot 10^{-104}\right):\\ \;\;\;\;x + \left(\left(\sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(y - z\right)} \cdot \sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(y - z\right)}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(y - z\right)}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right) - \frac{t \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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)))))