\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]

↓

\[\frac{v \cdot \log_* (1 + (e^{\frac{-t1}{t1 + u}} - 1)^*)}{t1 + u}\]

double f(double u, double v, double t1) {
        double r1985187 = t1;
        double r1985188 = -r1985187;
        double r1985189 = v;
        double r1985190 = r1985188 * r1985189;
        double r1985191 = u;
        double r1985192 = r1985187 + r1985191;
        double r1985193 = r1985192 * r1985192;
        double r1985194 = r1985190 / r1985193;
        return r1985194;
}

↓

double f(double u, double v, double t1) {
        double r1985195 = v;
        double r1985196 = t1;
        double r1985197 = -r1985196;
        double r1985198 = u;
        double r1985199 = r1985196 + r1985198;
        double r1985200 = r1985197 / r1985199;
        double r1985201 = expm1(r1985200);
        double r1985202 = log1p(r1985201);
        double r1985203 = r1985195 * r1985202;
        double r1985204 = r1985203 / r1985199;
        return r1985204;
}

\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}

↓

\frac{v \cdot \log_* (1 + (e^{\frac{-t1}{t1 + u}} - 1)^*)}{t1 + u}

Derivation

Initial program 18.2
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
Using strategy rm
Applied times-frac1.5
\[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}}\]
Using strategy rm
Applied associate-*r/1.3
\[\leadsto \color{blue}{\frac{\frac{-t1}{t1 + u} \cdot v}{t1 + u}}\]
Using strategy rm
Applied log1p-expm1-u1.3
\[\leadsto \frac{\color{blue}{\log_* (1 + (e^{\frac{-t1}{t1 + u}} - 1)^*)} \cdot v}{t1 + u}\]
Final simplification1.3
\[\leadsto \frac{v \cdot \log_* (1 + (e^{\frac{-t1}{t1 + u}} - 1)^*)}{t1 + u}\]

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))

Error

Derivation

Reproduce