Average Error: 2.1 → 1.6
Time: 3.3s
Precision: 64
\[\frac{x}{y} \cdot \left(z - t\right) + t\]
↓
\[\left(x \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{y}} + t\]
\frac{x}{y} \cdot \left(z - t\right) + t
↓
\left(x \cdot \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{y}} + t
double f(double x, double y, double z, double t) {
        double r409470 = x;
        double r409471 = y;
        double r409472 = r409470 / r409471;
        double r409473 = z;
        double r409474 = t;
        double r409475 = r409473 - r409474;
        double r409476 = r409472 * r409475;
        double r409477 = r409476 + r409474;
        return r409477;
}
↓
double f(double x, double y, double z, double t) {
        double r409478 = x;
        double r409479 = z;
        double r409480 = t;
        double r409481 = r409479 - r409480;
        double r409482 = cbrt(r409481);
        double r409483 = r409482 * r409482;
        double r409484 = y;
        double r409485 = cbrt(r409484);
        double r409486 = r409485 * r409485;
        double r409487 = r409483 / r409486;
        double r409488 = r409478 * r409487;
        double r409489 = r409482 / r409485;
        double r409490 = r409488 * r409489;
        double r409491 = r409490 + r409480;
        return r409491;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.1
Target2.3
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z \lt 2.7594565545626922 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z \lt 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array}\]

Derivation

  1. Initial program 2.1

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

  3. Applied div-inv2.1

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

  4. Applied associate-*l*6.6

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

  5. Simplified6.5

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

  7. Applied add-cube-cbrt7.0

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

  8. Applied add-cube-cbrt7.1

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

  9. Applied times-frac7.1

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

  10. Applied associate-*r*1.6

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

  11. Final simplification1.6

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

Reproduce

herbie shell --seed 2020024 +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))