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 r20596 = t1;
        double r20597 = -r20596;
        double r20598 = v;
        double r20599 = r20597 * r20598;
        double r20600 = u;
        double r20601 = r20596 + r20600;
        double r20602 = r20601 * r20601;
        double r20603 = r20599 / r20602;
        return r20603;
}
↓
double f(double u, double v, double t1) {
        double r20604 = v;
        double r20605 = t1;
        double r20606 = u;
        double r20607 = r20605 + r20606;
        double r20608 = r20604 / r20607;
        double r20609 = -1.0;
        double r20610 = r20609 / r20607;
        double r20611 = r20610 * r20605;
        double r20612 = expm1(r20611);
        double r20613 = log1p(r20612);
        double r20614 = r20608 * r20613;
        return r20614;
}

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))))