Average Error: 4.7 → 2.2
Time: 33.8s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\mathsf{fma}\left(y, z \cdot \left(\left(\left(\sqrt[3]{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)}\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)} - \tanh \left(\frac{x}{y}\right)\right) + \tanh \left(\frac{x}{y}\right) \cdot \left(\left(-1\right) + 1\right)\right), x\right)\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\mathsf{fma}\left(y, z \cdot \left(\left(\left(\sqrt[3]{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)}\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)} - \tanh \left(\frac{x}{y}\right)\right) + \tanh \left(\frac{x}{y}\right) \cdot \left(\left(-1\right) + 1\right)\right), x\right)
double f(double x, double y, double z, double t) {
        double r334291 = x;
        double r334292 = y;
        double r334293 = z;
        double r334294 = r334292 * r334293;
        double r334295 = t;
        double r334296 = r334295 / r334292;
        double r334297 = tanh(r334296);
        double r334298 = r334291 / r334292;
        double r334299 = tanh(r334298);
        double r334300 = r334297 - r334299;
        double r334301 = r334294 * r334300;
        double r334302 = r334291 + r334301;
        return r334302;
}
↓
double f(double x, double y, double z, double t) {
        double r334303 = y;
        double r334304 = z;
        double r334305 = t;
        double r334306 = r334305 / r334303;
        double r334307 = tanh(r334306);
        double r334308 = cbrt(r334307);
        double r334309 = r334308 * r334308;
        double r334310 = r334309 * r334308;
        double r334311 = x;
        double r334312 = r334311 / r334303;
        double r334313 = tanh(r334312);
        double r334314 = r334310 - r334313;
        double r334315 = 1.0;
        double r334316 = -r334315;
        double r334317 = r334316 + r334315;
        double r334318 = r334313 * r334317;
        double r334319 = r334314 + r334318;
        double r334320 = r334304 * r334319;
        double r334321 = fma(r334303, r334320, r334311);
        return r334321;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.7
Target2.1
Herbie2.2
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Initial program 4.7

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

  2. Simplified2.1

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

  4. Applied add-cube-cbrt2.1

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

  5. Applied add-cube-cbrt2.2

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

  6. Applied prod-diff2.2

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

  7. Simplified2.2

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

  8. Simplified2.2

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

  9. Final simplification2.2

    \[\leadsto \mathsf{fma}\left(y, z \cdot \left(\left(\left(\sqrt[3]{\tanh \left(\frac{t}{y}\right)} \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)}\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right)} - \tanh \left(\frac{x}{y}\right)\right) + \tanh \left(\frac{x}{y}\right) \cdot \left(\left(-1\right) + 1\right)\right), x\right)\]

Reproduce

herbie shell --seed 2020020 +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))))))