Average Error: 4.6 → 1.3
Time: 7.8s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\left(y \cdot \tanh \left(\frac{t}{y}\right) + y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z + x\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\left(y \cdot \tanh \left(\frac{t}{y}\right) + y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z + x
double f(double x, double y, double z, double t) {
        double r286616 = x;
        double r286617 = y;
        double r286618 = z;
        double r286619 = r286617 * r286618;
        double r286620 = t;
        double r286621 = r286620 / r286617;
        double r286622 = tanh(r286621);
        double r286623 = r286616 / r286617;
        double r286624 = tanh(r286623);
        double r286625 = r286622 - r286624;
        double r286626 = r286619 * r286625;
        double r286627 = r286616 + r286626;
        return r286627;
}
↓
double f(double x, double y, double z, double t) {
        double r286628 = y;
        double r286629 = t;
        double r286630 = r286629 / r286628;
        double r286631 = tanh(r286630);
        double r286632 = r286628 * r286631;
        double r286633 = x;
        double r286634 = r286633 / r286628;
        double r286635 = tanh(r286634);
        double r286636 = -r286635;
        double r286637 = r286628 * r286636;
        double r286638 = r286632 + r286637;
        double r286639 = z;
        double r286640 = r286638 * r286639;
        double r286641 = r286640 + r286633;
        return r286641;
}

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

Original4.6
Target1.8
Herbie1.3
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Initial program 4.6

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

  2. Simplified1.7

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

  4. Applied add-cube-cbrt2.1

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

  5. Applied associate-*l*2.1

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

  7. Applied fma-udef2.1

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

  8. Simplified1.3

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

  10. Applied sub-neg1.3

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

  11. Applied distribute-lft-in1.3

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

  12. Final simplification1.3

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

Reproduce

herbie shell --seed 2020036 +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))))))