Average Error: 4.5 → 2.2
Time: 7.4s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(z \cdot \left(\sqrt[3]{\sqrt[3]{y} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \left(\sqrt[3]{\sqrt[3]{y} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{y} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}\right)\right)\right)\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
x + \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(z \cdot \left(\sqrt[3]{\sqrt[3]{y} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \left(\sqrt[3]{\sqrt[3]{y} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{y} \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}\right)\right)\right)
(FPCore (x y z t)
 :precision binary64
 (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))
↓
(FPCore (x y z t)
 :precision binary64
 (+
  x
  (*
   (* (cbrt y) (cbrt y))
   (*
    z
    (*
     (cbrt (* (cbrt y) (- (tanh (/ t y)) (tanh (/ x y)))))
     (*
      (cbrt (* (cbrt y) (- (tanh (/ t y)) (tanh (/ x y)))))
      (cbrt (* (cbrt y) (- (tanh (/ t y)) (tanh (/ x y)))))))))))
double code(double x, double y, double z, double t) {
	return ((double) (x + ((double) (((double) (y * z)) * ((double) (((double) tanh((t / y))) - ((double) tanh((x / y)))))))));
}
↓
double code(double x, double y, double z, double t) {
	return ((double) (x + ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) * ((double) (z * ((double) (((double) cbrt(((double) (((double) cbrt(y)) * ((double) (((double) tanh((t / y))) - ((double) tanh((x / y))))))))) * ((double) (((double) cbrt(((double) (((double) cbrt(y)) * ((double) (((double) tanh((t / y))) - ((double) tanh((x / y))))))))) * ((double) cbrt(((double) (((double) cbrt(y)) * ((double) (((double) tanh((t / y))) - ((double) tanh((x / y)))))))))))))))))));
}

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
Herbie2.2
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Initial program Error: 4.5 bits

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

  2. SimplifiedError: 2.0 bits

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

  4. Applied add-cube-cbrtError: 2.3 bits

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

  5. Applied associate-*l*Error: 2.3 bits

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

  6. SimplifiedError: 2.0 bits

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

  8. Applied add-cube-cbrtError: 2.2 bits

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

  9. Final simplificationError: 2.2 bits

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

Reproduce

herbie shell --seed 2020204 
(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))))))