Average Error: 4.5 → 1.8
Time: 22.7s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[x + \left(\sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \left(\sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}\right)\right) \cdot z\]
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 \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \left(\sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)} \cdot \sqrt[3]{y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)}\right)\right) \cdot z
double f(double x, double y, double z, double t) {
        double r184855 = x;
        double r184856 = y;
        double r184857 = z;
        double r184858 = r184856 * r184857;
        double r184859 = t;
        double r184860 = r184859 / r184856;
        double r184861 = tanh(r184860);
        double r184862 = r184855 / r184856;
        double r184863 = tanh(r184862);
        double r184864 = r184861 - r184863;
        double r184865 = r184858 * r184864;
        double r184866 = r184855 + r184865;
        return r184866;
}
↓
double f(double x, double y, double z, double t) {
        double r184867 = x;
        double r184868 = y;
        double r184869 = t;
        double r184870 = r184869 / r184868;
        double r184871 = tanh(r184870);
        double r184872 = r184867 / r184868;
        double r184873 = tanh(r184872);
        double r184874 = r184871 - r184873;
        double r184875 = r184868 * r184874;
        double r184876 = cbrt(r184875);
        double r184877 = r184876 * r184876;
        double r184878 = r184876 * r184877;
        double r184879 = z;
        double r184880 = r184878 * r184879;
        double r184881 = r184867 + r184880;
        return r184881;
}

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
Target1.9
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 4.5

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

  2. Simplified4.5

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

  4. Applied fma-udef4.5

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

  5. Simplified1.9

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

  7. Applied associate-*r*1.4

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

  8. Simplified1.4

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

  10. Applied add-cube-cbrt1.8

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

  11. Final simplification1.8

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"

  :herbie-target
  (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))

  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))