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

Average Error: 2.1 → 1.1

Time: 4.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r649058 = x;
        double r649059 = y;
        double r649060 = r649058 / r649059;
        double r649061 = z;
        double r649062 = t;
        double r649063 = r649061 - r649062;
        double r649064 = r649060 * r649063;
        double r649065 = r649064 + r649062;
        return r649065;
}

↓

double f(double x, double y, double z, double t) {
        double r649066 = x;
        double r649067 = cbrt(r649066);
        double r649068 = r649067 * r649067;
        double r649069 = y;
        double r649070 = cbrt(r649069);
        double r649071 = r649070 * r649070;
        double r649072 = r649068 / r649071;
        double r649073 = cbrt(r649072);
        double r649074 = r649067 / r649070;
        double r649075 = cbrt(r649074);
        double r649076 = r649075 * r649075;
        double r649077 = r649073 * r649076;
        double r649078 = z;
        double r649079 = t;
        double r649080 = r649078 - r649079;
        double r649081 = r649074 * r649080;
        double r649082 = r649073 * r649081;
        double r649083 = r649077 * r649082;
        double r649084 = r649083 + r649079;
        return r649084;
}

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	1.1

\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

1. Initial program 2.1

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

3. Applied *-un-lft-identity 2.1

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

5. Applied add-cube-cbrt 2.7

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

6. Applied add-cube-cbrt 2.8

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

7. Applied times-frac 2.8

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

8. Applied associate-*l* 1.0

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

10. Applied add-cube-cbrt 1.1

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

11. Applied associate-*l* 1.1

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

13. Applied times-frac 1.1

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

14. Applied cbrt-prod 1.1

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

15. Final simplification 1.1

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

Reproduce

herbie shell --seed 2020035 
(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))