Average Error: 4.5 → 1.3
Time: 5.8s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
↓
\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\\ \mathbf{if}\;x + \left(y \cdot z\right) \cdot t_1 \leq 6.332555478161527 \cdot 10^{+305}:\\ \;\;\;\;x + y \cdot \left(z \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\begin{array}{l}
t_1 := \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\\
\mathbf{if}\;x + \left(y \cdot z\right) \cdot t_1 \leq 6.332555478161527 \cdot 10^{+305}:\\
\;\;\;\;x + y \cdot \left(z \cdot t_1\right)\\

\mathbf{else}:\\
\;\;\;\;z \cdot \left(t - x\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
 (let* ((t_1 (- (tanh (/ t y)) (tanh (/ x y)))))
   (if (<= (+ x (* (* y z) t_1)) 6.332555478161527e+305)
     (+ x (* y (* z t_1)))
     (* z (- t x)))))
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 t_1 = tanh(t / y) - tanh(x / y);
	double tmp;
	if ((x + ((y * z) * t_1)) <= 6.332555478161527e+305) {
		tmp = x + (y * (z * t_1));
	} else {
		tmp = z * (t - x);
	}
	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

Original4.5
Target2.0
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 2 regimes

  2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 6.33255547816152672e305

    1. Initial program 2.1

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

    3. Applied associate-*l*_binary641.1

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

    if 6.33255547816152672e305 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

    1. Initial program 59.5

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

    2. Taylor expanded around inf 6.5

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

    3. Simplified6.5

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

    4. Taylor expanded around inf 7.2

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

    5. Simplified7.2

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \leq 6.332555478161527 \cdot 10^{+305}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(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))))))