Average Error: 17.7 → 1.2
Time: 17.5s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{v}{t1 + u} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{t1 + u} \cdot t1\right)\right)\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{v}{t1 + u} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{t1 + u} \cdot t1\right)\right)
double f(double u, double v, double t1) {
        double r20580 = t1;
        double r20581 = -r20580;
        double r20582 = v;
        double r20583 = r20581 * r20582;
        double r20584 = u;
        double r20585 = r20580 + r20584;
        double r20586 = r20585 * r20585;
        double r20587 = r20583 / r20586;
        return r20587;
}
↓
double f(double u, double v, double t1) {
        double r20588 = v;
        double r20589 = t1;
        double r20590 = u;
        double r20591 = r20589 + r20590;
        double r20592 = r20588 / r20591;
        double r20593 = -1.0;
        double r20594 = r20593 / r20591;
        double r20595 = r20594 * r20589;
        double r20596 = expm1(r20595);
        double r20597 = log1p(r20596);
        double r20598 = r20592 * r20597;
        return r20598;
}

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.7

    \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
  2. Using strategy rm

  3. Applied times-frac1.2

    \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}}\]
  4. Using strategy rm

  5. Applied log1p-expm1-u1.2

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-t1}{t1 + u}\right)\right)} \cdot \frac{v}{t1 + u}\]
  6. Using strategy rm

  7. Applied div-inv1.2

    \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(-t1\right) \cdot \frac{1}{t1 + u}}\right)\right) \cdot \frac{v}{t1 + u}\]

  8. Final simplification1.2

    \[\leadsto \frac{v}{t1 + u} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-1}{t1 + u} \cdot t1\right)\right)\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))