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

Average Error: 2.0 → 1.4

Time: 4.6s

Precision: binary64

Math

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

↓

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

C

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x / y)) * ((double) (z - t)))) + t));
}

↓

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / ((double) (((double) cbrt(y)) * ((double) cbrt(y)))))) * ((double) (((double) (((double) cbrt(x)) / ((double) cbrt(y)))) * ((double) (((double) cbrt(((double) (z - t)))) * ((double) cbrt(((double) (z - t)))))))))) * ((double) cbrt(((double) (z - t)))))) + t));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	2.0
Target	2.3
Herbie	1.4

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

Derivation

1. Initial program 2.0

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

3. Applied add-cube-cbrt 2.5

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

4. Applied associate-*r* 2.5

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

6. Applied add-cube-cbrt 2.6

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

7. Applied add-cube-cbrt 2.7

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

8. Applied times-frac 2.7

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

9. Applied associate-*l* 1.4

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

10. Final simplification 1.4

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

Reproduce

herbie shell --seed 2020162 
(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))