Average Error: 4.5 → 1.6
Time: 44.4s
Precision: binary64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]
↓
\[x + z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
x + z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\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 (* z (* y (- (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) {
	return x + (z * (y * (tanh(t / y) - 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
Target2.1
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. Initial program 4.5

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

  2. Applied sub-neg_binary644.5

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

  3. Applied distribute-rgt-in_binary644.5

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

  4. Applied add-cube-cbrt_binary644.9

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

  5. Simplified4.9

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

  6. Simplified2.0

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

  7. Applied pow1/3_binary6424.9

    \[\leadsto x + \left(\sqrt[3]{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} \cdot \sqrt[3]{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)}\right) \cdot \color{blue}{{\left(z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}^{0.3333333333333333}} \]

  8. Applied pow1/3_binary6425.1

    \[\leadsto x + \left(\sqrt[3]{z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)} \cdot \color{blue}{{\left(z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}^{0.3333333333333333}}\right) \cdot {\left(z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}^{0.3333333333333333} \]

  9. Applied pow1/3_binary6425.2

    \[\leadsto x + \left(\color{blue}{{\left(z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}^{0.3333333333333333}} \cdot {\left(z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}^{0.3333333333333333}\right) \cdot {\left(z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}^{0.3333333333333333} \]

  10. Applied pow-sqr_binary6425.2

    \[\leadsto x + \color{blue}{{\left(z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}^{\left(2 \cdot 0.3333333333333333\right)}} \cdot {\left(z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}^{0.3333333333333333} \]

  11. Applied pow-prod-up_binary641.6

    \[\leadsto x + \color{blue}{{\left(z \cdot \left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\right)}^{\left(2 \cdot 0.3333333333333333 + 0.3333333333333333\right)}} \]

  12. Final simplification1.6

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

Reproduce

herbie shell --seed 2022077 
(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))))))