Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1

Average Error: 2.1 → 1.6

Time: 20.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1237315393794427982429025852194816 \lor \neg \left(y \le 3.262590146382767270401479675683423871414 \cdot 10^{-223}\right):\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z - t}{\frac{\sqrt[3]{y}}{\sqrt[3]{x}}} + \mathsf{fma}\left(\left(-t\right) + t, \frac{x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot x}{y} + t\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r323507 = x;
        double r323508 = y;
        double r323509 = r323507 / r323508;
        double r323510 = z;
        double r323511 = t;
        double r323512 = r323510 - r323511;
        double r323513 = r323509 * r323512;
        double r323514 = r323513 + r323511;
        return r323514;
}

↓

double f(double x, double y, double z, double t) {
        double r323515 = y;
        double r323516 = -1.237315393794428e+33;
        bool r323517 = r323515 <= r323516;
        double r323518 = 3.2625901463827673e-223;
        bool r323519 = r323515 <= r323518;
        double r323520 = !r323519;
        bool r323521 = r323517 || r323520;
        double r323522 = x;
        double r323523 = cbrt(r323522);
        double r323524 = r323523 * r323523;
        double r323525 = cbrt(r323515);
        double r323526 = r323525 * r323525;
        double r323527 = r323524 / r323526;
        double r323528 = z;
        double r323529 = t;
        double r323530 = r323528 - r323529;
        double r323531 = r323525 / r323523;
        double r323532 = r323530 / r323531;
        double r323533 = r323527 * r323532;
        double r323534 = -r323529;
        double r323535 = r323534 + r323529;
        double r323536 = r323522 / r323515;
        double r323537 = fma(r323535, r323536, r323529);
        double r323538 = r323533 + r323537;
        double r323539 = r323530 * r323522;
        double r323540 = r323539 / r323515;
        double r323541 = r323540 + r323529;
        double r323542 = r323521 ? r323538 : r323541;
        return r323542;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.1
Target	2.3
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;z \lt 2.759456554562692182563154937894909044548 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.32699445087443595687739933019129648094 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -1.237315393794428e+33 or 3.2625901463827673e-223 < y

   1. Initial program 1.5

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

   3. Applied *-un-lft-identity 1.5

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

   4. Applied add-cube-cbrt 1.8

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

   5. Applied prod-diff 1.8

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

   6. Applied distribute-lft-in 1.8

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

   7. Applied associate-+l+ 1.8

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

   8. Simplified 1.8

      \[\leadsto \frac{x}{y} \cdot \mathsf{fma}\left(\sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z}, -t \cdot 1\right) + \color{blue}{\mathsf{fma}\left(\left(-t\right) + t, \frac{x}{y}, t\right)}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 2.0

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

   11. Applied add-cube-cbrt 2.1

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

   12. Applied times-frac 2.1

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

   13. Applied associate-*l* 1.4

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

   14. Simplified 1.3

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

   if -1.237315393794428e+33 < y < 3.2625901463827673e-223

   1. Initial program 3.8

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

   3. Applied associate-*l/ 2.6

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

   4. Simplified 2.6

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1237315393794427982429025852194816 \lor \neg \left(y \le 3.262590146382767270401479675683423871414 \cdot 10^{-223}\right):\\ \;\;\;\;\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z - t}{\frac{\sqrt[3]{y}}{\sqrt[3]{x}}} + \mathsf{fma}\left(\left(-t\right) + t, \frac{x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z - t\right) \cdot x}{y} + t\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

  (+ (* (/ x y) (- z t)) t))