SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.9 → 2.2

Time: 54.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r513932 = x;
        double r513933 = y;
        double r513934 = z;
        double r513935 = r513933 * r513934;
        double r513936 = t;
        double r513937 = r513936 / r513933;
        double r513938 = tanh(r513937);
        double r513939 = r513932 / r513933;
        double r513940 = tanh(r513939);
        double r513941 = r513938 - r513940;
        double r513942 = r513935 * r513941;
        double r513943 = r513932 + r513942;
        return r513943;
}

↓

double f(double x, double y, double z, double t) {
        double r513944 = y;
        double r513945 = z;
        double r513946 = t;
        double r513947 = r513946 / r513944;
        double r513948 = tanh(r513947);
        double r513949 = cbrt(r513948);
        double r513950 = r513949 * r513949;
        double r513951 = x;
        double r513952 = r513951 / r513944;
        double r513953 = tanh(r513952);
        double r513954 = 1.0;
        double r513955 = r513953 * r513954;
        double r513956 = -r513955;
        double r513957 = fma(r513950, r513949, r513956);
        double r513958 = r513945 * r513957;
        double r513959 = -r513954;
        double r513960 = r513959 + r513954;
        double r513961 = r513953 * r513960;
        double r513962 = r513945 * r513961;
        double r513963 = r513958 + r513962;
        double r513964 = fma(r513944, r513963, r513951);
        return r513964;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.9
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. Initial program 4.9

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

2. Simplified 2.1

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

4. Applied add-cube-cbrt 2.1

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

5. Applied add-cube-cbrt 2.2

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

6. Applied prod-diff 2.2

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

7. Applied distribute-lft-in 2.2

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

8. Simplified 2.2

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

9. Simplified 2.2

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

10. Final simplification 2.2

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

Reproduce

herbie shell --seed 2020025 +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))))))