Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.9 → 8.6

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{t - x}{a - z} \cdot \left(y - z\right) \le -2.335375383468043 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{\frac{y - z}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}} + x\\ \mathbf{elif}\;x + \frac{t - x}{a - z} \cdot \left(y - z\right) \le 0.0:\\ \;\;\;\;\left(t + \frac{x \cdot y}{z}\right) - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}}{\sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right) \cdot \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}}{\sqrt[3]{a - z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r7054356 = x;
        double r7054357 = y;
        double r7054358 = z;
        double r7054359 = r7054357 - r7054358;
        double r7054360 = t;
        double r7054361 = r7054360 - r7054356;
        double r7054362 = a;
        double r7054363 = r7054362 - r7054358;
        double r7054364 = r7054361 / r7054363;
        double r7054365 = r7054359 * r7054364;
        double r7054366 = r7054356 + r7054365;
        return r7054366;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r7054367 = x;
        double r7054368 = t;
        double r7054369 = r7054368 - r7054367;
        double r7054370 = a;
        double r7054371 = z;
        double r7054372 = r7054370 - r7054371;
        double r7054373 = r7054369 / r7054372;
        double r7054374 = y;
        double r7054375 = r7054374 - r7054371;
        double r7054376 = r7054373 * r7054375;
        double r7054377 = r7054367 + r7054376;
        double r7054378 = -2.335375383468043e-253;
        bool r7054379 = r7054377 <= r7054378;
        double r7054380 = cbrt(r7054372);
        double r7054381 = r7054380 * r7054380;
        double r7054382 = cbrt(r7054381);
        double r7054383 = r7054375 / r7054382;
        double r7054384 = r7054383 / r7054381;
        double r7054385 = cbrt(r7054382);
        double r7054386 = r7054384 / r7054385;
        double r7054387 = cbrt(r7054380);
        double r7054388 = cbrt(r7054387);
        double r7054389 = r7054369 / r7054388;
        double r7054390 = r7054386 * r7054389;
        double r7054391 = r7054390 + r7054367;
        double r7054392 = 0.0;
        bool r7054393 = r7054377 <= r7054392;
        double r7054394 = r7054367 * r7054374;
        double r7054395 = r7054394 / r7054371;
        double r7054396 = r7054368 + r7054395;
        double r7054397 = r7054374 * r7054368;
        double r7054398 = r7054397 / r7054371;
        double r7054399 = r7054396 - r7054398;
        double r7054400 = cbrt(r7054375);
        double r7054401 = sqrt(r7054382);
        double r7054402 = r7054400 / r7054401;
        double r7054403 = r7054402 / r7054380;
        double r7054404 = r7054369 / r7054387;
        double r7054405 = r7054403 * r7054404;
        double r7054406 = r7054400 * r7054400;
        double r7054407 = r7054406 / r7054401;
        double r7054408 = r7054407 / r7054380;
        double r7054409 = r7054405 * r7054408;
        double r7054410 = r7054367 + r7054409;
        double r7054411 = r7054393 ? r7054399 : r7054410;
        double r7054412 = r7054379 ? r7054391 : r7054411;
        return r7054412;
}

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 (+ x (* (- y z) (/ (- t x) (- a z)))) < -2.335375383468043e-253

   1. Initial program 7.0

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

   3. Applied add-cube-cbrt 7.6

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

   4. Applied *-un-lft-identity 7.6

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

   5. Applied times-frac 7.7

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

   6. Applied associate-*r* 5.4

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

   7. Simplified 5.4

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

   9. Applied add-cube-cbrt 5.4

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

   10. Applied cbrt-prod 5.5

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

   11. Applied *-un-lft-identity 5.5

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

   12. Applied times-frac 5.5

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

   13. Applied associate-*r* 5.5

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

   14. Simplified 5.5

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

   16. Applied add-cube-cbrt 5.5

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

   17. Applied cbrt-prod 5.5

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

   18. Applied cbrt-prod 5.6

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

   19. Applied *-un-lft-identity 5.6

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

   20. Applied times-frac 5.6

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

   21. Applied associate-*r* 5.5

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

   22. Simplified 5.5

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

   if -2.335375383468043e-253 < (+ x (* (- y z) (/ (- t x) (- a z)))) < 0.0

   1. Initial program 59.4

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

   2. Taylor expanded around inf 29.0

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

   if 0.0 < (+ x (* (- y z) (/ (- t x) (- a z))))

   1. Initial program 7.7

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

   3. Applied add-cube-cbrt 8.4

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

   4. Applied *-un-lft-identity 8.4

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

   5. Applied times-frac 8.4

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

   6. Applied associate-*r* 5.5

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

   7. Simplified 5.5

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

   9. Applied add-cube-cbrt 5.5

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

   10. Applied cbrt-prod 5.5

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

   11. Applied *-un-lft-identity 5.5

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

   12. Applied times-frac 5.6

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

   13. Applied associate-*r* 5.2

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

   14. Simplified 5.2

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

   16. Applied add-sqr-sqrt 5.2

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

   17. Applied add-cube-cbrt 5.1

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

   18. Applied times-frac 5.2

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

   19. Applied times-frac 5.2

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

   20. Applied associate-*l* 4.8

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

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{t - x}{a - z} \cdot \left(y - z\right) \le -2.335375383468043 \cdot 10^{-253}:\\ \;\;\;\;\frac{\frac{\frac{y - z}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{\sqrt[3]{a - z}}}} + x\\ \mathbf{elif}\;x + \frac{t - x}{a - z} \cdot \left(y - z\right) \le 0.0:\\ \;\;\;\;\left(t + \frac{x \cdot y}{z}\right) - \frac{y \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}}{\sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right) \cdot \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}}}{\sqrt[3]{a - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  (+ x (* (- y z) (/ (- t x) (- a z)))))