Average Error: 4.6 → 1.3
Time: 6.9s
Precision: 64
\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\]
↓
\[\left(y \cdot \tanh \left(\frac{t}{y}\right) + y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z + x\]
x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)
↓
\left(y \cdot \tanh \left(\frac{t}{y}\right) + y \cdot \left(-\tanh \left(\frac{x}{y}\right)\right)\right) \cdot z + x
double f(double x, double y, double z, double t) {
        double r278032 = x;
        double r278033 = y;
        double r278034 = z;
        double r278035 = r278033 * r278034;
        double r278036 = t;
        double r278037 = r278036 / r278033;
        double r278038 = tanh(r278037);
        double r278039 = r278032 / r278033;
        double r278040 = tanh(r278039);
        double r278041 = r278038 - r278040;
        double r278042 = r278035 * r278041;
        double r278043 = r278032 + r278042;
        return r278043;
}
↓
double f(double x, double y, double z, double t) {
        double r278044 = y;
        double r278045 = t;
        double r278046 = r278045 / r278044;
        double r278047 = tanh(r278046);
        double r278048 = r278044 * r278047;
        double r278049 = x;
        double r278050 = r278049 / r278044;
        double r278051 = tanh(r278050);
        double r278052 = -r278051;
        double r278053 = r278044 * r278052;
        double r278054 = r278048 + r278053;
        double r278055 = z;
        double r278056 = r278054 * r278055;
        double r278057 = r278056 + r278049;
        return r278057;
}

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
Herbie1.3
\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right)\]

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. Simplified1.7

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

  4. Applied add-cube-cbrt2.1

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

  5. Applied associate-*l*2.1

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

  7. Applied fma-udef2.1

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

  8. Simplified1.3

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

  10. Applied sub-neg1.3

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

  11. Applied distribute-lft-in1.3

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

  12. Final simplification1.3

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

Reproduce

herbie shell --seed 2020036 +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))))))