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

Average Error: 1.9 → 1.7

Time: 25.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.091870269310474233912169562668515221046 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;t + \left(\frac{x}{y} \cdot \left(\left(-t\right) + t\right) + \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z - t}{\frac{\sqrt[3]{y}}{\sqrt[3]{x}}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r289137 = x;
        double r289138 = y;
        double r289139 = r289137 / r289138;
        double r289140 = z;
        double r289141 = t;
        double r289142 = r289140 - r289141;
        double r289143 = r289139 * r289142;
        double r289144 = r289143 + r289141;
        return r289144;
}

↓

double f(double x, double y, double z, double t) {
        double r289145 = x;
        double r289146 = -4.091870269310474e-74;
        bool r289147 = r289145 <= r289146;
        double r289148 = z;
        double r289149 = t;
        double r289150 = r289148 - r289149;
        double r289151 = y;
        double r289152 = r289150 / r289151;
        double r289153 = r289145 * r289152;
        double r289154 = r289153 + r289149;
        double r289155 = r289145 / r289151;
        double r289156 = -r289149;
        double r289157 = r289156 + r289149;
        double r289158 = r289155 * r289157;
        double r289159 = cbrt(r289145);
        double r289160 = r289159 * r289159;
        double r289161 = cbrt(r289151);
        double r289162 = r289161 * r289161;
        double r289163 = r289160 / r289162;
        double r289164 = r289161 / r289159;
        double r289165 = r289150 / r289164;
        double r289166 = r289163 * r289165;
        double r289167 = r289158 + r289166;
        double r289168 = r289149 + r289167;
        double r289169 = r289147 ? r289154 : r289168;
        return r289169;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.9
Target	2.3
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if x < -4.091870269310474e-74

   1. Initial program 3.1

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

   3. Applied div-inv 3.1

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

   4. Applied associate-*l* 2.5

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

   5. Simplified 2.4

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

   if -4.091870269310474e-74 < x

   1. Initial program 1.5

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

   3. Applied add-cube-cbrt 2.0

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

   4. Applied add-cube-cbrt 2.1

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

   5. Applied times-frac 2.1

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

   6. Applied associate-*l* 0.8

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

   8. Applied add-cube-cbrt 0.8

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

   9. Applied add-sqr-sqrt 31.7

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

   10. Applied prod-diff 31.7

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

   11. Applied distribute-lft-in 31.7

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

   12. Applied distribute-lft-in 31.7

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

   13. Simplified 1.5

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

   14. Simplified 1.5

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

   16. Applied *-un-lft-identity 1.5

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

   17. Applied *-un-lft-identity 1.5

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

   18. Applied times-frac 1.5

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

   19. Applied associate-*l* 1.5

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

   20. Simplified 1.5

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

   22. Applied add-cube-cbrt 2.0

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

   23. Applied add-cube-cbrt 2.1

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

   24. Applied times-frac 2.1

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

   25. Applied *-un-lft-identity 2.1

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

   26. Applied times-frac 1.5

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

   27. Simplified 1.4

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.091870269310474233912169562668515221046 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;t + \left(\frac{x}{y} \cdot \left(\left(-t\right) + t\right) + \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z - t}{\frac{\sqrt[3]{y}}{\sqrt[3]{x}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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))