SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.7 → 2.0

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.1600489093627797 \cdot 10^{137}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \tanh \left(\frac{t}{y}\right) + z \cdot \left(-\tanh \left(\frac{x}{y}\right)\right), x\right)\\ \end{array}\]

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 ((y <= -1.1600489093627797e+137)) {
		VAR = fma(z, t, (x - (x * z)));
	} else {
		VAR = fma(y, ((z * tanh((t / y))) + (z * -tanh((x / y)))), x);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.7
Target	2.1
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. Split input into 2 regimes

2. if y < -1.1600489093627797e+137

   1. Initial program 15.8

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

   3. Applied sub-neg 15.8

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

   4. Applied distribute-lft-in 15.8

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

   5. Applied associate-+r+ 16.0

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

   6. Simplified 15.9

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

   7. Taylor expanded around inf 7.1

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

   8. Simplified 7.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, x - x \cdot z\right)}\]

   if -1.1600489093627797e+137 < y

   1. Initial program 3.1

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

   2. Simplified 1.2

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

   4. Applied sub-neg 1.2

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

   5. Applied distribute-lft-in 1.2

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.1600489093627797 \cdot 10^{137}:\\ \;\;\;\;\mathsf{fma}\left(z, t, x - x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z \cdot \tanh \left(\frac{t}{y}\right) + z \cdot \left(-\tanh \left(\frac{x}{y}\right)\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(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))))))