Average Error: 4.9 → 2.4
Time: 11.1s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
↓
\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\\ \mathbf{if}\;x + \left(y \cdot z\right) \cdot t_1 \leq 4.889942237596147 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, t_1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]
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 4.889942237596147 \cdot 10^{+303}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot z, t_1, x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(z, t - x, 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)) 4.889942237596147e+303)
     (fma (* y z) t_1 x)
     (fma z (- t x) 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)) <= 4.889942237596147e+303) {
		tmp = fma((y * z), t_1, x);
	} else {
		tmp = fma(z, (t - x), x);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original4.9
Target1.9
Herbie2.4
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \]

Derivation

  1. Split input into 2 regimes

  2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 4.8899422375961473e303

    1. Initial program 2.2

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

    2. Simplified2.2

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

    if 4.8899422375961473e303 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Taylor expanded in y around inf 6.2

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

    4. Simplified6.2

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

    5. Taylor expanded in z around 0 6.2

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

    6. Simplified6.2

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \leq 4.889942237596147 \cdot 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot z, \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t - x, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))))))