Average Error: 17.7 → 1.2
Time: 17.6s
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 r20561 = t1;
        double r20562 = -r20561;
        double r20563 = v;
        double r20564 = r20562 * r20563;
        double r20565 = u;
        double r20566 = r20561 + r20565;
        double r20567 = r20566 * r20566;
        double r20568 = r20564 / r20567;
        return r20568;
}
↓
double f(double u, double v, double t1) {
        double r20569 = v;
        double r20570 = t1;
        double r20571 = u;
        double r20572 = r20570 + r20571;
        double r20573 = r20569 / r20572;
        double r20574 = -1.0;
        double r20575 = r20574 / r20572;
        double r20576 = r20575 * r20570;
        double r20577 = expm1(r20576);
        double r20578 = log1p(r20577);
        double r20579 = r20573 * r20578;
        return r20579;
}

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))))