Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.6 → 11.5

Time: 32.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.409051721433192 \cdot 10^{178} \lor \neg \left(z \le 1.20516532875887789 \cdot 10^{156}\right):\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(t - x\right)\right) \cdot \frac{1}{\sqrt[3]{a - z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r142391 = x;
        double r142392 = y;
        double r142393 = z;
        double r142394 = r142392 - r142393;
        double r142395 = t;
        double r142396 = r142395 - r142391;
        double r142397 = a;
        double r142398 = r142397 - r142393;
        double r142399 = r142396 / r142398;
        double r142400 = r142394 * r142399;
        double r142401 = r142391 + r142400;
        return r142401;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r142402 = z;
        double r142403 = -1.409051721433192e+178;
        bool r142404 = r142402 <= r142403;
        double r142405 = 1.2051653287588779e+156;
        bool r142406 = r142402 <= r142405;
        double r142407 = !r142406;
        bool r142408 = r142404 || r142407;
        double r142409 = y;
        double r142410 = x;
        double r142411 = r142410 / r142402;
        double r142412 = t;
        double r142413 = r142412 / r142402;
        double r142414 = r142411 - r142413;
        double r142415 = r142409 * r142414;
        double r142416 = r142415 + r142412;
        double r142417 = r142409 - r142402;
        double r142418 = a;
        double r142419 = r142418 - r142402;
        double r142420 = cbrt(r142419);
        double r142421 = r142420 * r142420;
        double r142422 = r142417 / r142421;
        double r142423 = r142412 - r142410;
        double r142424 = r142422 * r142423;
        double r142425 = 1.0;
        double r142426 = r142425 / r142420;
        double r142427 = r142424 * r142426;
        double r142428 = r142410 + r142427;
        double r142429 = r142408 ? r142416 : r142428;
        return r142429;
}

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 < -1.409051721433192e+178 or 1.2051653287588779e+156 < z

   1. Initial program 27.6

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

   3. Applied add-cube-cbrt 28.1

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

   4. Applied *-un-lft-identity 28.1

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

   5. Applied times-frac 28.1

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

   6. Applied associate-*r* 23.4

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

   7. Simplified 23.4

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

   9. Applied add-cube-cbrt 23.6

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

   10. Applied associate-*r* 23.6

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

   11. Taylor expanded around inf 25.3

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

   12. Simplified 16.0

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

   if -1.409051721433192e+178 < z < 1.2051653287588779e+156

   1. Initial program 10.2

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

   3. Applied add-cube-cbrt 10.7

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

   4. Applied *-un-lft-identity 10.7

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

   5. Applied times-frac 10.7

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

   6. Applied associate-*r* 8.5

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

   7. Simplified 8.5

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

   9. Applied div-inv 8.5

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

   10. Applied associate-*r* 10.0

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

   11. Simplified 10.0

       \[\leadsto x + \color{blue}{\left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(t - x\right)\right)} \cdot \frac{1}{\sqrt[3]{a - z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.409051721433192 \cdot 10^{178} \lor \neg \left(z \le 1.20516532875887789 \cdot 10^{156}\right):\\ \;\;\;\;y \cdot \left(\frac{x}{z} - \frac{t}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \left(t - x\right)\right) \cdot \frac{1}{\sqrt[3]{a - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  (+ x (* (- y z) (/ (- t x) (- a z)))))