Average Error: 4.4 → 1.6
Time: 5.2s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[x + \left(\left(y \cdot \tanh \left(\frac{t}{y}\right)\right) \cdot z + \left(y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z\right)\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
x + \left(\left(y \cdot \tanh \left(\frac{t}{y}\right)\right) \cdot z + \left(y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z\right)
double f(double x, double y, double z, double t) {
        double r388389 = x;
        double r388390 = y;
        double r388391 = z;
        double r388392 = r388390 * r388391;
        double r388393 = t;
        double r388394 = r388393 / r388390;
        double r388395 = tanh(r388394);
        double r388396 = r388389 / r388390;
        double r388397 = tanh(r388396);
        double r388398 = r388395 - r388397;
        double r388399 = r388392 * r388398;
        double r388400 = r388389 + r388399;
        return r388400;
}
↓
double f(double x, double y, double z, double t) {
        double r388401 = x;
        double r388402 = y;
        double r388403 = t;
        double r388404 = r388403 / r388402;
        double r388405 = tanh(r388404);
        double r388406 = r388402 * r388405;
        double r388407 = z;
        double r388408 = r388406 * r388407;
        double r388409 = r388401 / r388402;
        double r388410 = tanh(r388409);
        double r388411 = -r388410;
        double r388412 = r388402 * r388411;
        double r388413 = r388412 * r388407;
        double r388414 = r388408 + r388413;
        double r388415 = r388401 + r388414;
        return r388415;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original4.4
Target2.0
Herbie1.6
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Initial program 4.4

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

  3. Applied associate-*l*2.0

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

  5. Applied add-cbrt-cube5.0

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

  6. Simplified5.0

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

  8. Applied sub-neg5.0

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

  9. Applied distribute-lft-in5.0

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

  10. Applied distribute-lft-in5.1

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

  11. Simplified1.6

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

  12. Simplified1.6

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

  14. Applied associate-*r*1.6

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

  15. Final simplification1.6

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

Reproduce

herbie shell --seed 2020027 
(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))))))