Average Error: 4.9 → 1.7
Time: 13.8s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq 6.498537120547806 \cdot 10^{+217}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z \cdot t - x \cdot z\right)\\ \end{array}\]
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}\;y \leq 6.498537120547806 \cdot 10^{+217}:\\
\;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\

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

\end{array}
(FPCore (x y z t)
 :precision binary64
 (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))
↓
(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.498537120547806e+217)
   (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))
   (+ x (- (* z t) (* x z)))))
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 tmp;
	if (y <= 6.498537120547806e+217) {
		tmp = x + (y * (z * (tanh(t / y) - tanh(x / y))));
	} else {
		tmp = x + ((z * t) - (x * z));
	}
	return tmp;
}

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

Original4.9
Target2.2
Herbie1.7
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if y < 6.4985371205478062e217

    1. Initial program 3.8

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

    3. Applied associate-*l*_binary64_85441.5

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

    if 6.4985371205478062e217 < y

    1. Initial program 21.8

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

    2. Taylor expanded around inf 4.1

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

    3. Simplified4.1

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

    4. Taylor expanded around 0 4.1

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.498537120547806 \cdot 10^{+217}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(z \cdot t - x \cdot z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021047 
(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))))))