Numeric.Signal:interpolate from hsignal-0.2.7.1

Average Error: 14.7 → 6.9

Time: 16.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -1.3363846895601392 \cdot 10^{-286}:\\ \;\;\;\;x + \left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\left(\sqrt[3]{y - z} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)\right) \cdot \frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 3.24883698955848775 \cdot 10^{-308}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r124601 = x;
        double r124602 = y;
        double r124603 = z;
        double r124604 = r124602 - r124603;
        double r124605 = t;
        double r124606 = r124605 - r124601;
        double r124607 = a;
        double r124608 = r124607 - r124603;
        double r124609 = r124606 / r124608;
        double r124610 = r124604 * r124609;
        double r124611 = r124601 + r124610;
        return r124611;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r124612 = x;
        double r124613 = y;
        double r124614 = z;
        double r124615 = r124613 - r124614;
        double r124616 = t;
        double r124617 = r124616 - r124612;
        double r124618 = a;
        double r124619 = r124618 - r124614;
        double r124620 = r124617 / r124619;
        double r124621 = r124615 * r124620;
        double r124622 = r124612 + r124621;
        double r124623 = -1.3363846895601392e-286;
        bool r124624 = r124622 <= r124623;
        double r124625 = cbrt(r124615);
        double r124626 = r124625 * r124625;
        double r124627 = cbrt(r124617);
        double r124628 = r124627 * r124627;
        double r124629 = r124625 * r124628;
        double r124630 = 1.0;
        double r124631 = cbrt(r124619);
        double r124632 = r124631 * r124631;
        double r124633 = r124630 / r124632;
        double r124634 = r124629 * r124633;
        double r124635 = r124626 * r124634;
        double r124636 = r124627 / r124631;
        double r124637 = r124635 * r124636;
        double r124638 = r124612 + r124637;
        double r124639 = 3.2488369895584877e-308;
        bool r124640 = r124622 <= r124639;
        double r124641 = r124612 / r124614;
        double r124642 = r124616 / r124614;
        double r124643 = r124641 - r124642;
        double r124644 = r124613 * r124643;
        double r124645 = r124616 + r124644;
        double r124646 = r124628 / r124632;
        double r124647 = r124625 * r124646;
        double r124648 = r124626 * r124647;
        double r124649 = r124648 * r124636;
        double r124650 = r124612 + r124649;
        double r124651 = r124640 ? r124645 : r124650;
        double r124652 = r124624 ? r124638 : r124651;
        return r124652;
}

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)))) < -1.3363846895601392e-286

   1. Initial program 7.2

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

   3. Applied add-cube-cbrt 7.9

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

   4. Applied add-cube-cbrt 8.1

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

   5. Applied times-frac 8.1

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

   6. Applied associate-*r* 4.8

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

   8. Applied add-cube-cbrt 4.8

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

   9. Applied associate-*l* 4.8

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

   11. Applied div-inv 4.8

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

   12. Applied associate-*r* 4.4

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

   if -1.3363846895601392e-286 < (+ x (* (- y z) (/ (- t x) (- a z)))) < 3.2488369895584877e-308

   1. Initial program 60.7

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

   3. Applied add-cube-cbrt 60.4

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

   4. Applied add-cube-cbrt 60.4

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

   5. Applied times-frac 60.4

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

   6. Applied associate-*r* 59.5

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

   8. Applied add-cube-cbrt 59.5

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

   9. Applied associate-*l* 59.4

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

   11. Applied div-inv 59.5

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

   12. Applied associate-*r* 59.3

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

   13. Taylor expanded around inf 27.1

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

   14. Simplified 21.3

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

   if 3.2488369895584877e-308 < (+ x (* (- y z) (/ (- t x) (- a z))))

   1. Initial program 7.5

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

   3. Applied add-cube-cbrt 8.2

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

   4. Applied add-cube-cbrt 8.4

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

   5. Applied times-frac 8.4

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

   6. Applied associate-*r* 4.7

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

   8. Applied add-cube-cbrt 4.7

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

   9. Applied associate-*l* 4.7

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

4. Final simplification 6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le -1.3363846895601392 \cdot 10^{-286}:\\ \;\;\;\;x + \left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\left(\sqrt[3]{y - z} \cdot \left(\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}\right)\right) \cdot \frac{1}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \le 3.24883698955848775 \cdot 10^{-308}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \left(\sqrt[3]{y - z} \cdot \frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}\right)\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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)))))