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 r19025097 = x;
        double r19025098 = y;
        double r19025099 = r19025097 / r19025098;
        double r19025100 = z;
        double r19025101 = t;
        double r19025102 = r19025100 - r19025101;
        double r19025103 = r19025099 * r19025102;
        double r19025104 = r19025103 + r19025101;
        return r19025104;
}

↓

double f(double x, double y, double z, double t) {
        double r19025105 = z;
        double r19025106 = t;
        double r19025107 = r19025105 - r19025106;
        double r19025108 = cbrt(r19025107);
        double r19025109 = y;
        double r19025110 = x;
        double r19025111 = cbrt(r19025110);
        double r19025112 = r19025109 / r19025111;
        double r19025113 = r19025108 / r19025112;
        double r19025114 = r19025111 * r19025111;
        double r19025115 = r19025108 * r19025108;
        double r19025116 = r19025114 * r19025115;
        double r19025117 = r19025113 * r19025116;
        double r19025118 = r19025117 + r19025106;
        return r19025118;
}

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))