SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.8 → 2.2

Time: 26.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot \left(\sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}\right)\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}, x\right)\\ \mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 1.972480810362636817765665532393837592151 \cdot 10^{305}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r199209 = x;
        double r199210 = y;
        double r199211 = z;
        double r199212 = r199210 * r199211;
        double r199213 = t;
        double r199214 = r199213 / r199210;
        double r199215 = tanh(r199214);
        double r199216 = r199209 / r199210;
        double r199217 = tanh(r199216);
        double r199218 = r199215 - r199217;
        double r199219 = r199212 * r199218;
        double r199220 = r199209 + r199219;
        return r199220;
}

↓

double f(double x, double y, double z, double t) {
        double r199221 = x;
        double r199222 = y;
        double r199223 = z;
        double r199224 = r199222 * r199223;
        double r199225 = t;
        double r199226 = r199225 / r199222;
        double r199227 = tanh(r199226);
        double r199228 = r199221 / r199222;
        double r199229 = tanh(r199228);
        double r199230 = r199227 - r199229;
        double r199231 = r199224 * r199230;
        double r199232 = r199221 + r199231;
        double r199233 = -inf.0;
        bool r199234 = r199232 <= r199233;
        double r199235 = cbrt(r199230);
        double r199236 = r199235 * r199235;
        double r199237 = r199223 * r199236;
        double r199238 = r199237 * r199235;
        double r199239 = fma(r199222, r199238, r199221);
        double r199240 = 1.972480810362637e+305;
        bool r199241 = r199232 <= r199240;
        double r199242 = r199221 * r199223;
        double r199243 = r199221 - r199242;
        double r199244 = r199241 ? r199232 : r199243;
        double r199245 = r199234 ? r199239 : r199244;
        return r199245;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.8
Target	2.1
Herbie	2.2

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 2.3

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

   4. Applied add-cube-cbrt 3.3

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

   5. Applied associate-*r* 3.3

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

   if -inf.0 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < 1.972480810362637e+305

   1. Initial program 0.6

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

   if 1.972480810362637e+305 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y)))))

   1. Initial program 59.4

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

   2. Simplified 21.1

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

   3. Taylor expanded around inf 33.1

      \[\leadsto \color{blue}{x - x \cdot z}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot \left(\sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}\right)\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)}, x\right)\\ \mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 1.972480810362636817765665532393837592151 \cdot 10^{305}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot z\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +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))))))