SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.6 → 1.8

Time: 6.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty \lor \neg \left(x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 9.69418639124574765 \cdot 10^{292}\right):\\ \;\;\;\;x + \left(\left(\sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)} \cdot \sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)}\right) \cdot \sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)} + y \cdot \left(\left(-\tanh \left(\frac{x}{y}\right)\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \end{array}\]

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) {
	double VAR;
	if ((((x + ((y * z) * (tanh((t / y)) - tanh((x / y))))) <= -inf.0) || !((x + ((y * z) * (tanh((t / y)) - tanh((x / y))))) <= 9.694186391245748e+292))) {
		VAR = (x + (((cbrt((y * (tanh((t / y)) * z))) * cbrt((y * (tanh((t / y)) * z)))) * cbrt((y * (tanh((t / y)) * z)))) + (y * (-tanh((x / y)) * z))));
	} else {
		VAR = (x + ((y * z) * (tanh((t / y)) - tanh((x / y)))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.6
Target	2.0
Herbie	1.8

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

Derivation

1. Split input into 2 regimes

2. if (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < -inf.0 or 9.694186391245748e+292 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y)))))

   1. Initial program 52.7

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

   3. Applied associate-*l* 13.1

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

   5. Applied sub-neg 13.1

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

   6. Applied distribute-lft-in 13.1

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

   7. Applied distribute-lft-in 15.4

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

   8. Simplified 15.4

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

   9. Simplified 15.4

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

   11. Applied add-cube-cbrt 15.7

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

   if -inf.0 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < 9.694186391245748e+292

   1. Initial program 0.6

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty \lor \neg \left(x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 9.69418639124574765 \cdot 10^{292}\right):\\ \;\;\;\;x + \left(\left(\sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)} \cdot \sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)}\right) \cdot \sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot z\right)} + y \cdot \left(\left(-\tanh \left(\frac{x}{y}\right)\right) \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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))))))