Average Error: 4.3 → 1.3
Time: 8.5s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;x + \left(\left(\tanh \left(\frac{t}{y}\right) \cdot z\right) \cdot y + \left(-x\right) \cdot z\right)\\ \mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 2.3928587330146112 \cdot 10^{295}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t \cdot z + \left(y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z\right)\\ \end{array}\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\begin{array}{l}
\mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\
\;\;\;\;x + \left(\left(\tanh \left(\frac{t}{y}\right) \cdot z\right) \cdot y + \left(-x\right) \cdot z\right)\\

\mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 2.3928587330146112 \cdot 10^{295}:\\
\;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + \left(t \cdot z + \left(y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z\right)\\

\end{array}
double f(double x, double y, double z, double t) {
        double r430481 = x;
        double r430482 = y;
        double r430483 = z;
        double r430484 = r430482 * r430483;
        double r430485 = t;
        double r430486 = r430485 / r430482;
        double r430487 = tanh(r430486);
        double r430488 = r430481 / r430482;
        double r430489 = tanh(r430488);
        double r430490 = r430487 - r430489;
        double r430491 = r430484 * r430490;
        double r430492 = r430481 + r430491;
        return r430492;
}
↓
double f(double x, double y, double z, double t) {
        double r430493 = x;
        double r430494 = y;
        double r430495 = z;
        double r430496 = r430494 * r430495;
        double r430497 = t;
        double r430498 = r430497 / r430494;
        double r430499 = tanh(r430498);
        double r430500 = r430493 / r430494;
        double r430501 = tanh(r430500);
        double r430502 = r430499 - r430501;
        double r430503 = r430496 * r430502;
        double r430504 = r430493 + r430503;
        double r430505 = -inf.0;
        bool r430506 = r430504 <= r430505;
        double r430507 = r430499 * r430495;
        double r430508 = r430507 * r430494;
        double r430509 = -r430493;
        double r430510 = r430509 * r430495;
        double r430511 = r430508 + r430510;
        double r430512 = r430493 + r430511;
        double r430513 = 2.3928587330146112e+295;
        bool r430514 = r430504 <= r430513;
        double r430515 = r430497 * r430495;
        double r430516 = -r430501;
        double r430517 = r430494 * r430516;
        double r430518 = r430517 * r430495;
        double r430519 = r430515 + r430518;
        double r430520 = r430493 + r430519;
        double r430521 = r430514 ? r430504 : r430520;
        double r430522 = r430506 ? r430512 : r430521;
        return r430522;
}

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

Original4.3
Target2.1
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. Split input into 3 regimes

  2. if (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < -inf.0

    1. Initial program 64.0

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

    3. Applied associate-*l*1.3

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

    5. Applied sub-neg1.3

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

    6. Applied distribute-lft-in1.3

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

    7. Applied distribute-lft-in1.3

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

    8. Simplified1.3

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

    9. Simplified1.3

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

    11. Applied associate-*r*1.2

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

    12. Taylor expanded around 0 0.9

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

    13. Simplified0.9

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

    if -inf.0 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))) < 2.3928587330146112e+295

    1. Initial program 0.6

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

    if 2.3928587330146112e+295 < (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y)))))

    1. Initial program 47.4

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

    3. Applied associate-*l*18.8

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

    5. Applied sub-neg18.8

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

    6. Applied distribute-lft-in18.8

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

    7. Applied distribute-lft-in21.5

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

    8. Simplified21.5

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

    9. Simplified21.5

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

    11. Applied associate-*r*21.4

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

    12. Taylor expanded around 0 15.6

      \[\leadsto x + \left(\color{blue}{t \cdot z} + \left(y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) = -\infty:\\ \;\;\;\;x + \left(\left(\tanh \left(\frac{t}{y}\right) \cdot z\right) \cdot y + \left(-x\right) \cdot z\right)\\ \mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \le 2.3928587330146112 \cdot 10^{295}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(t \cdot z + \left(y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))))