SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.4 → 3.9

Time: 6.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.00905380261296578 \cdot 10^{47} \lor \neg \left(y \le 45574721038.576889\right):\\ \;\;\;\;x + \left(\left({\left(\sqrt[3]{y}\right)}^{3} \cdot \tanh \left(\frac{t}{y}\right)\right) \cdot z + -1 \cdot \left(x \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \sqrt[3]{{\left(\tanh \left(\frac{x}{y}\right)\right)}^{3}}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((y <= -2.0090538026129658e+47) || !(y <= 45574721038.57689))) {
		VAR = ((double) (x + ((double) (((double) (((double) (((double) pow(((double) cbrt(y)), 3.0)) * ((double) tanh(((double) (t / y)))))) * z)) + ((double) (-1.0 * ((double) (x * z))))))));
	} else {
		VAR = ((double) (x + ((double) (((double) (y * z)) * ((double) (((double) tanh(((double) (t / y)))) - ((double) cbrt(((double) pow(((double) tanh(((double) (x / y)))), 3.0))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.4
Target	1.9
Herbie	3.9

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

Derivation

1. Split input into 2 regimes

2. if y < -2.00905380261296578e47 or 45574721038.576889 < y

   1. Initial program 10.8

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

   3. Applied associate-*l* 4.6

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

   5. Applied add-cube-cbrt 5.0

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

   6. Applied associate-*l* 5.1

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

   8. Applied sub-neg 5.1

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

   9. Applied distribute-lft-in 5.1

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

   10. Applied distribute-lft-in 5.2

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

   11. Applied distribute-lft-in 5.5

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

   12. Simplified 4.6

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

   13. Simplified 4.6

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

   14. Taylor expanded around 0 8.5

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

   if -2.00905380261296578e47 < y < 45574721038.576889

   1. Initial program 0.1

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

   3. Applied add-cbrt-cube 0.7

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

   4. Simplified 0.7

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

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.00905380261296578 \cdot 10^{47} \lor \neg \left(y \le 45574721038.576889\right):\\ \;\;\;\;x + \left(\left({\left(\sqrt[3]{y}\right)}^{3} \cdot \tanh \left(\frac{t}{y}\right)\right) \cdot z + -1 \cdot \left(x \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \sqrt[3]{{\left(\tanh \left(\frac{x}{y}\right)\right)}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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))))))