Average Error: 4.7 → 2.5
Time: 18.4s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[x + \left(\sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \left(\sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot y\right)\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) \cdot z + \left(-\tanh \left(\frac{x}{y}\right)\right) \cdot z}\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
x + \left(\sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \left(\sqrt[3]{z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot y\right)\right) \cdot \sqrt[3]{\tanh \left(\frac{t}{y}\right) \cdot z + \left(-\tanh \left(\frac{x}{y}\right)\right) \cdot z}
double f(double x, double y, double z, double t) {
        double r198748 = x;
        double r198749 = y;
        double r198750 = z;
        double r198751 = r198749 * r198750;
        double r198752 = t;
        double r198753 = r198752 / r198749;
        double r198754 = tanh(r198753);
        double r198755 = r198748 / r198749;
        double r198756 = tanh(r198755);
        double r198757 = r198754 - r198756;
        double r198758 = r198751 * r198757;
        double r198759 = r198748 + r198758;
        return r198759;
}
↓
double f(double x, double y, double z, double t) {
        double r198760 = x;
        double r198761 = z;
        double r198762 = t;
        double r198763 = y;
        double r198764 = r198762 / r198763;
        double r198765 = tanh(r198764);
        double r198766 = r198760 / r198763;
        double r198767 = tanh(r198766);
        double r198768 = r198765 - r198767;
        double r198769 = r198761 * r198768;
        double r198770 = cbrt(r198769);
        double r198771 = r198770 * r198763;
        double r198772 = r198770 * r198771;
        double r198773 = r198765 * r198761;
        double r198774 = -r198767;
        double r198775 = r198774 * r198761;
        double r198776 = r198773 + r198775;
        double r198777 = cbrt(r198776);
        double r198778 = r198772 * r198777;
        double r198779 = r198760 + r198778;
        return r198779;
}

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

Original4.7
Target2.2
Herbie2.5
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

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. Using strategy rm

  3. Applied associate-*l*2.2

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

  5. Applied sub-neg2.2

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

  6. Applied distribute-lft-in2.2

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

  7. Simplified2.2

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

  8. Simplified2.2

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

  10. Applied add-cube-cbrt2.5

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

  11. Applied associate-*r*2.5

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

  12. Simplified2.5

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

  13. Final simplification2.5

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

Reproduce

herbie shell --seed 2019325 
(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))))))