SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.5 → 1.7

Time: 52.2s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r342799 = x;
        double r342800 = y;
        double r342801 = z;
        double r342802 = r342800 * r342801;
        double r342803 = t;
        double r342804 = r342803 / r342800;
        double r342805 = tanh(r342804);
        double r342806 = r342799 / r342800;
        double r342807 = tanh(r342806);
        double r342808 = r342805 - r342807;
        double r342809 = r342802 * r342808;
        double r342810 = r342799 + r342809;
        return r342810;
}

↓

double f(double x, double y, double z, double t) {
        double r342811 = x;
        double r342812 = z;
        double r342813 = t;
        double r342814 = y;
        double r342815 = r342813 / r342814;
        double r342816 = tanh(r342815);
        double r342817 = r342816 * r342814;
        double r342818 = r342812 * r342817;
        double r342819 = r342811 / r342814;
        double r342820 = tanh(r342819);
        double r342821 = -r342820;
        double r342822 = r342821 * r342814;
        double r342823 = r342812 * r342822;
        double r342824 = r342818 + r342823;
        double r342825 = r342811 + r342824;
        return r342825;
}

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	1.7

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

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 2.3

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

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 sub-neg 2.4

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

9. Applied distribute-lft-in 2.4

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

10. Applied distribute-lft-in 2.4

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

11. Applied distribute-lft-in 2.5

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

12. Simplified 3.0

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

13. Simplified 1.7

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

14. Final simplification 1.7

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

Reproduce

herbie shell --seed 2019208 
(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))))))