Average Error: 4.6 → 1.8
Time: 5.6s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\mathsf{fma}\left(y, z \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\tanh \left(\frac{t}{y}\right)\right)\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\mathsf{fma}\left(y, z \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\tanh \left(\frac{t}{y}\right)\right)\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)
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) {
	return fma(y, (z * (log1p(expm1(tanh((t / y)))) - tanh((x / y)))), x);
}

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

Original4.6
Target1.8
Herbie1.8
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Initial program 4.6

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

  2. Simplified1.8

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

  4. Applied log1p-expm1-u1.8

    \[\leadsto \mathsf{fma}\left(y, z \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tanh \left(\frac{t}{y}\right)\right)\right)} - \tanh \left(\frac{x}{y}\right)\right), x\right)\]

  5. Final simplification1.8

    \[\leadsto \mathsf{fma}\left(y, z \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\tanh \left(\frac{t}{y}\right)\right)\right) - \tanh \left(\frac{x}{y}\right)\right), x\right)\]

Reproduce

herbie shell --seed 2020092 +o rules:numerics
(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))))))