Average Error: 4.8 → 1.4
Time: 6.6s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \cdot z + x\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \cdot z + x
double f(double x, double y, double z, double t) {
        double r259683 = x;
        double r259684 = y;
        double r259685 = z;
        double r259686 = r259684 * r259685;
        double r259687 = t;
        double r259688 = r259687 / r259684;
        double r259689 = tanh(r259688);
        double r259690 = r259683 / r259684;
        double r259691 = tanh(r259690);
        double r259692 = r259689 - r259691;
        double r259693 = r259686 * r259692;
        double r259694 = r259683 + r259693;
        return r259694;
}
↓
double f(double x, double y, double z, double t) {
        double r259695 = y;
        double r259696 = t;
        double r259697 = r259696 / r259695;
        double r259698 = tanh(r259697);
        double r259699 = x;
        double r259700 = r259699 / r259695;
        double r259701 = tanh(r259700);
        double r259702 = r259698 - r259701;
        double r259703 = r259695 * r259702;
        double r259704 = z;
        double r259705 = r259703 * r259704;
        double r259706 = r259705 + r259699;
        return r259706;
}

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

Original4.8
Target2.0
Herbie1.4
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

Derivation

  1. Initial program 4.8

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

  2. Simplified2.0

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

  4. Applied add-cube-cbrt2.3

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

  5. Applied associate-*l*2.3

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

  7. Applied fma-udef2.3

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

  8. Simplified1.4

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

  9. Final simplification1.4

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

Reproduce

herbie shell --seed 2020081 +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))))))