Average Error: 4.5 → 1.6
Time: 15.3s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.1428028320336448 \cdot 10^{+179}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array}\]
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 -1.1428028320336448 \cdot 10^{+179}:\\
\;\;\;\;x + z \cdot \left(t - x\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\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 -1.1428028320336448e+179)
   (+ x (* z (- t x)))
   (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))))
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 <= -1.1428028320336448e+179) {
		tmp = x + (z * (t - x));
	} else {
		tmp = x + (y * (z * (tanh(t / y) - tanh(x / y))));
	}
	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
Herbie1.6
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1.14280283203364483e179

    1. Initial program 18.8

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

    3. Applied associate-*l*_binary649.0

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

    5. Applied add-cube-cbrt_binary649.4

      \[\leadsto x + y \cdot \left(\color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

    6. Applied associate-*l*_binary649.4

      \[\leadsto x + y \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\sqrt[3]{z} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}\]

    7. Simplified9.4

      \[\leadsto x + y \cdot \left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \color{blue}{\left(\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot \sqrt[3]{z}\right)}\right)\]
    8. Using strategy rm

    9. Applied flip--_binary6426.1

      \[\leadsto x + y \cdot \left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \left(\color{blue}{\frac{\tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right) \cdot \tanh \left(\frac{x}{y}\right)}{\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)}} \cdot \sqrt[3]{z}\right)\right)\]

    10. Applied associate-*l/_binary6426.4

      \[\leadsto x + y \cdot \left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \color{blue}{\frac{\left(\tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right) \cdot \tanh \left(\frac{x}{y}\right)\right) \cdot \sqrt[3]{z}}{\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)}}\right)\]

    11. Simplified26.4

      \[\leadsto x + y \cdot \left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \frac{\color{blue}{\sqrt[3]{z} \cdot \left(\tanh \left(\frac{t}{y}\right) \cdot \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right) \cdot \tanh \left(\frac{x}{y}\right)\right)}}{\tanh \left(\frac{t}{y}\right) + \tanh \left(\frac{x}{y}\right)}\right)\]

    12. Taylor expanded around 0 4.6

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

    13. Simplified4.6

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

    if -1.14280283203364483e179 < y

    1. Initial program 3.1

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

    3. Applied associate-*l*_binary641.3

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

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1428028320336448 \cdot 10^{+179}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(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))))))