Average Error: 4.8 → 1.2
Time: 8.4s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
↓
\[\begin{array}{l} t_1 := x + z \cdot \left(t - x\right)\\ t_2 := \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\\ t_3 := x + \left(y \cdot z\right) \cdot t_2\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq 2.381352279137968 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, t_2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
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 := x + z \cdot \left(t - x\right)\\
t_2 := \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\\
t_3 := x + \left(y \cdot z\right) \cdot t_2\\
\mathbf{if}\;t_3 \leq -\infty:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_3 \leq 2.381352279137968 \cdot 10^{+295}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot z, t_2, x\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\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 (+ x (* z (- t x))))
        (t_2 (- (tanh (/ t y)) (tanh (/ x y))))
        (t_3 (+ x (* (* y z) t_2))))
   (if (<= t_3 (- INFINITY))
     t_1
     (if (<= t_3 2.381352279137968e+295) (fma (* y z) t_2 x) t_1))))
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 = x + (z * (t - x));
	double t_2 = tanh((t / y)) - tanh((x / y));
	double t_3 = x + ((y * z) * t_2);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_3 <= 2.381352279137968e+295) {
		tmp = fma((y * z), t_2, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.8
Target2.1
Herbie1.2
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \]

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))))) < -inf.0 or 2.381352279137968e295 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

    1. Initial program 52.9

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

    2. Simplified52.9

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

    3. Taylor expanded in y around inf 8.0

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

    4. Simplified8.0

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

    if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 2.381352279137968e295

    1. Initial program 0.6

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

    2. Simplified0.6

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \leq -\infty:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \leq 2.381352279137968 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))))))