Average Error: 4.6 → 0.8
Time: 5.9s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
↓
\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\\ \mathbf{if}\;\begin{array}{l} t_2 := x + \left(y \cdot z\right) \cdot t_1\\ t_2 \leq -\infty \lor \neg \left(t_2 \leq 1.0953066441175457 \cdot 10^{+308}\right) \end{array}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, t_1, x\right)\\ \end{array} \]
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}\;\begin{array}{l}
t_2 := x + \left(y \cdot z\right) \cdot t_1\\
t_2 \leq -\infty \lor \neg \left(t_2 \leq 1.0953066441175457 \cdot 10^{+308}\right)
\end{array}:\\
\;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot z, t_1, 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 (let* ((t_2 (+ x (* (* y z) t_1))))
         (or (<= t_2 (- INFINITY)) (not (<= t_2 1.0953066441175457e+308))))
     (fma (- t x) z x)
     (fma (* y z) t_1 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 t_2 = x + ((y * z) * t_1);
	double tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 1.0953066441175457e+308)) {
		tmp = fma((t - x), z, x);
	} else {
		tmp = fma((y * z), t_1, x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.6
Target2.0
Herbie0.8
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \]

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 1.09530664411754572e308 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

    1. Initial program 63.9

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

    2. Simplified63.9

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

    3. Applied add-log-exp_binary6463.9

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

    4. Taylor expanded in y around inf 3.7

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

    5. Simplified3.7

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

    6. Taylor expanded in x around 0 3.7

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

    7. Simplified3.7

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

    if -inf.0 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 1.09530664411754572e308

    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 simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \leq -\infty \lor \neg \left(x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \leq 1.0953066441175457 \cdot 10^{+308}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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))))))