Average Error: 14.6 → 11.3
Time: 30.6s
Precision: 64
\[x + \left(y - z\right) \cdot \frac{t - x}{a - z}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -4.461695944389990584855395905754140013375 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \mathbf{elif}\;a \le 2.213657479282627496070255771891102626534 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \end{array}\]
x + \left(y - z\right) \cdot \frac{t - x}{a - z}
↓
\begin{array}{l}
\mathbf{if}\;a \le -4.461695944389990584855395905754140013375 \cdot 10^{-130}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\

\mathbf{elif}\;a \le 2.213657479282627496070255771891102626534 \cdot 10^{-92}:\\
\;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a) {
        double r11859903 = x;
        double r11859904 = y;
        double r11859905 = z;
        double r11859906 = r11859904 - r11859905;
        double r11859907 = t;
        double r11859908 = r11859907 - r11859903;
        double r11859909 = a;
        double r11859910 = r11859909 - r11859905;
        double r11859911 = r11859908 / r11859910;
        double r11859912 = r11859906 * r11859911;
        double r11859913 = r11859903 + r11859912;
        return r11859913;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r11859914 = a;
        double r11859915 = -4.4616959443899906e-130;
        bool r11859916 = r11859914 <= r11859915;
        double r11859917 = x;
        double r11859918 = cbrt(r11859917);
        double r11859919 = r11859918 * r11859918;
        double r11859920 = y;
        double r11859921 = z;
        double r11859922 = r11859920 - r11859921;
        double r11859923 = r11859914 - r11859921;
        double r11859924 = cbrt(r11859923);
        double r11859925 = r11859924 * r11859924;
        double r11859926 = r11859922 / r11859925;
        double r11859927 = cbrt(r11859925);
        double r11859928 = r11859926 / r11859927;
        double r11859929 = t;
        double r11859930 = r11859929 - r11859917;
        double r11859931 = cbrt(r11859924);
        double r11859932 = r11859930 / r11859931;
        double r11859933 = r11859928 * r11859932;
        double r11859934 = fma(r11859919, r11859918, r11859933);
        double r11859935 = 2.2136574792826275e-92;
        bool r11859936 = r11859914 <= r11859935;
        double r11859937 = r11859917 / r11859921;
        double r11859938 = r11859929 * r11859920;
        double r11859939 = r11859938 / r11859921;
        double r11859940 = r11859929 - r11859939;
        double r11859941 = fma(r11859937, r11859920, r11859940);
        double r11859942 = r11859936 ? r11859941 : r11859934;
        double r11859943 = r11859916 ? r11859934 : r11859942;
        return r11859943;
}

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 < -4.4616959443899906e-130 or 2.2136574792826275e-92 < 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-cbrt11.3

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

    4. Applied *-un-lft-identity11.3

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

    5. Applied times-frac11.3

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

    6. Applied associate-*r*9.2

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

    7. Simplified9.2

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

    9. Applied add-cube-cbrt9.7

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

    10. Applied fma-def9.7

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

    12. Applied add-cube-cbrt9.7

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

    13. Applied cbrt-prod9.8

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

    14. Applied *-un-lft-identity9.8

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

    15. Applied times-frac9.8

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

    16. Applied associate-*r*9.8

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

    17. Simplified9.8

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

    if -4.4616959443899906e-130 < a < 2.2136574792826275e-92

    1. Initial program 24.3

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

    2. Taylor expanded around inf 15.5

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

    3. Simplified15.1

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

  4. Final simplification11.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -4.461695944389990584855395905754140013375 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \mathbf{elif}\;a \le 2.213657479282627496070255771891102626534 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t - \frac{t \cdot y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x} \cdot \sqrt[3]{x}, \sqrt[3]{x}, \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 +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)))))