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

Average Error: 2.1 → 1.6

Time: 13.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r20674932 = x;
        double r20674933 = y;
        double r20674934 = r20674932 / r20674933;
        double r20674935 = z;
        double r20674936 = t;
        double r20674937 = r20674935 - r20674936;
        double r20674938 = r20674934 * r20674937;
        double r20674939 = r20674938 + r20674936;
        return r20674939;
}

↓

double f(double x, double y, double z, double t) {
        double r20674940 = z;
        double r20674941 = t;
        double r20674942 = r20674940 - r20674941;
        double r20674943 = cbrt(r20674942);
        double r20674944 = y;
        double r20674945 = cbrt(r20674944);
        double r20674946 = r20674943 / r20674945;
        double r20674947 = x;
        double r20674948 = r20674946 * r20674947;
        double r20674949 = r20674946 * r20674946;
        double r20674950 = r20674948 * r20674949;
        double r20674951 = -1.0;
        double r20674952 = fma(r20674941, r20674951, r20674941);
        double r20674953 = r20674952 * r20674947;
        double r20674954 = r20674953 / r20674944;
        double r20674955 = r20674950 + r20674954;
        double r20674956 = r20674955 + r20674941;
        return r20674956;
}

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. Initial program 2.1

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

3. Applied add-cube-cbrt 2.2

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

4. Applied add-sqr-sqrt 32.8

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

5. Applied prod-diff 32.8

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

6. Applied distribute-lft-in 32.8

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

7. Simplified 2.1

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

8. Simplified 2.0

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

10. Applied *-un-lft-identity 2.0

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

11. Applied add-cube-cbrt 2.6

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

12. Applied times-frac 2.6

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

13. Applied add-cube-cbrt 2.7

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

14. Applied times-frac 1.8

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

15. Simplified 1.9

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

16. Simplified 1.6

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

17. Final simplification 1.6

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"

  :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))