SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.5 → 2.0

Time: 6.8s

Precision: 64

Math

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

↓

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

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.5
Target	2.0
Herbie	2.0

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

Derivation

1. Initial program 4.5

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

3. Applied associate-*l* 2.0

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

5. Applied add-cube-cbrt 2.4

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

6. Applied associate-*l* 2.4

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

8. Applied pow1 2.4

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

9. Applied pow1 2.4

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

10. Applied pow-prod-down 2.4

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

11. Applied pow1 2.4

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

12. Applied pow-prod-down 2.4

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

13. Applied pow1 2.4

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

14. Applied pow1 2.4

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

15. Applied pow-prod-down 2.4

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

16. Applied pow-prod-down 2.4

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

17. Simplified 2.0

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

18. Final simplification 2.0

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

Reproduce

herbie shell --seed 2020058 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :herbie-target
  (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))

  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))