SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.1 → 1.4

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(z \cdot y\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) + x\\ \mathbf{elif}\;x + \left(z \cdot y\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 8.124775333408145 \cdot 10^{-45}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) + x\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r18089336 = x;
        double r18089337 = y;
        double r18089338 = z;
        double r18089339 = r18089337 * r18089338;
        double r18089340 = t;
        double r18089341 = r18089340 / r18089337;
        double r18089342 = tanh(r18089341);
        double r18089343 = r18089336 / r18089337;
        double r18089344 = tanh(r18089343);
        double r18089345 = r18089342 - r18089344;
        double r18089346 = r18089339 * r18089345;
        double r18089347 = r18089336 + r18089346;
        return r18089347;
}

↓

double f(double x, double y, double z, double t) {
        double r18089348 = x;
        double r18089349 = z;
        double r18089350 = y;
        double r18089351 = r18089349 * r18089350;
        double r18089352 = t;
        double r18089353 = r18089352 / r18089350;
        double r18089354 = tanh(r18089353);
        double r18089355 = r18089348 / r18089350;
        double r18089356 = tanh(r18089355);
        double r18089357 = r18089354 - r18089356;
        double r18089358 = r18089351 * r18089357;
        double r18089359 = r18089348 + r18089358;
        double r18089360 = -inf.0;
        bool r18089361 = r18089359 <= r18089360;
        double r18089362 = r18089349 * r18089357;
        double r18089363 = r18089350 * r18089362;
        double r18089364 = r18089363 + r18089348;
        double r18089365 = 8.124775333408145e-45;
        bool r18089366 = r18089359 <= r18089365;
        double r18089367 = r18089366 ? r18089359 : r18089364;
        double r18089368 = r18089361 ? r18089364 : r18089367;
        return r18089368;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.1
Target	1.9
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < -inf.0 or 8.124775333408145e-45 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y)))))

   1. Initial program 9.4

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

   3. Applied associate-*l* 2.3

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

   if -inf.0 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < 8.124775333408145e-45

   1. Initial program 0.8

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(z \cdot y\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) + x\\ \mathbf{elif}\;x + \left(z \cdot y\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 8.124775333408145 \cdot 10^{-45}:\\ \;\;\;\;x + \left(z \cdot y\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"

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

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