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

Average Error: 2.1 → 1.8

Time: 23.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r20359843 = x;
        double r20359844 = y;
        double r20359845 = r20359843 / r20359844;
        double r20359846 = z;
        double r20359847 = t;
        double r20359848 = r20359846 - r20359847;
        double r20359849 = r20359845 * r20359848;
        double r20359850 = r20359849 + r20359847;
        return r20359850;
}

↓

double f(double x, double y, double z, double t) {
        double r20359851 = z;
        double r20359852 = t;
        double r20359853 = r20359851 - r20359852;
        double r20359854 = cbrt(r20359853);
        double r20359855 = y;
        double r20359856 = x;
        double r20359857 = cbrt(r20359856);
        double r20359858 = r20359855 / r20359857;
        double r20359859 = r20359854 / r20359858;
        double r20359860 = r20359857 * r20359857;
        double r20359861 = r20359854 * r20359854;
        double r20359862 = r20359860 * r20359861;
        double r20359863 = r20359859 * r20359862;
        double r20359864 = r20359863 + r20359852;
        return r20359864;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.1
Target	2.4
Herbie	1.8

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

Derivation

1. Initial program 2.1

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

2. Taylor expanded around 0 6.4

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

3. Simplified 2.0

   \[\leadsto \color{blue}{\frac{z - t}{\frac{y}{x}}} + t\]
4. Using strategy rm

5. Applied add-cube-cbrt 2.5

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

6. Applied *-un-lft-identity 2.5

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

7. Applied times-frac 2.5

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

8. Applied add-cube-cbrt 2.6

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

9. Applied times-frac 1.8

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

10. Simplified 1.8

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

11. Final simplification 1.8

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

Reproduce

herbie shell --seed 2019172 +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))