Average Error: 4.5 → 5.4
Time: 1.1m
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.14779953698363994 \cdot 10^{156} \lor \neg \left(y \le 1.60579202525289377 \cdot 10^{127}\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot \left(y \cdot \left({\left(\tanh \left(\frac{t}{y}\right)\right)}^{3} - {\left(\tanh \left(\frac{x}{y}\right)\right)}^{3}\right)\right)}{\tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{t}{y}\right) + \left(\tanh \left(\frac{x}{y}\right) \cdot \tanh \left(\frac{x}{y}\right) + \tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{x}{y}\right)\right)}\\ \end{array}\]
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}\;y \le -2.14779953698363994 \cdot 10^{156} \lor \neg \left(y \le 1.60579202525289377 \cdot 10^{127}\right):\\
\;\;\;\;x + z \cdot \left(t - x\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{z \cdot \left(y \cdot \left({\left(\tanh \left(\frac{t}{y}\right)\right)}^{3} - {\left(\tanh \left(\frac{x}{y}\right)\right)}^{3}\right)\right)}{\tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{t}{y}\right) + \left(\tanh \left(\frac{x}{y}\right) \cdot \tanh \left(\frac{x}{y}\right) + \tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{x}{y}\right)\right)}\\

\end{array}
double code(double x, double y, double z, double t) {
	return (x + ((y * z) * (tanh((t / y)) - tanh((x / y)))));
}
↓
double code(double x, double y, double z, double t) {
	double VAR;
	if (((y <= -2.14779953698364e+156) || !(y <= 1.6057920252528938e+127))) {
		VAR = (x + (z * (t - x)));
	} else {
		VAR = (x + ((z * (y * (pow(tanh((t / y)), 3.0) - pow(tanh((x / y)), 3.0)))) / ((tanh((t / y)) * tanh((t / y))) + ((tanh((x / y)) * tanh((x / y))) + (tanh((t / y)) * tanh((x / y)))))));
	}
	return VAR;
}

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

Original4.5
Target2.1
Herbie5.4
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if y < -2.14779953698364e+156 or 1.6057920252528938e+127 < y

    1. Initial program 15.7

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

    3. Applied *-commutative15.7

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

    4. Applied associate-*l*5.7

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

    5. Taylor expanded around 0 6.9

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

    if -2.14779953698364e+156 < y < 1.6057920252528938e+127

    1. Initial program 1.0

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

    3. Applied *-commutative1.0

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

    4. Applied associate-*l*0.3

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

    6. Applied flip3--4.4

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

    7. Applied associate-*r/4.7

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

    8. Applied associate-*r/4.9

      \[\leadsto x + \color{blue}{\frac{z \cdot \left(y \cdot \left({\left(\tanh \left(\frac{t}{y}\right)\right)}^{3} - {\left(\tanh \left(\frac{x}{y}\right)\right)}^{3}\right)\right)}{\tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{t}{y}\right) + \left(\tanh \left(\frac{x}{y}\right) \cdot \tanh \left(\frac{x}{y}\right) + \tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{x}{y}\right)\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.14779953698363994 \cdot 10^{156} \lor \neg \left(y \le 1.60579202525289377 \cdot 10^{127}\right):\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot \left(y \cdot \left({\left(\tanh \left(\frac{t}{y}\right)\right)}^{3} - {\left(\tanh \left(\frac{x}{y}\right)\right)}^{3}\right)\right)}{\tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{t}{y}\right) + \left(\tanh \left(\frac{x}{y}\right) \cdot \tanh \left(\frac{x}{y}\right) + \tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{x}{y}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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))))))