Average Error: 18.5 → 1.6
Time: 22.5s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}
double f(double u, double v, double t1) {
        double r23563 = t1;
        double r23564 = -r23563;
        double r23565 = v;
        double r23566 = r23564 * r23565;
        double r23567 = u;
        double r23568 = r23563 + r23567;
        double r23569 = r23568 * r23568;
        double r23570 = r23566 / r23569;
        return r23570;
}
↓
double f(double u, double v, double t1) {
        double r23571 = t1;
        double r23572 = -r23571;
        double r23573 = u;
        double r23574 = r23571 + r23573;
        double r23575 = v;
        double r23576 = r23574 / r23575;
        double r23577 = r23572 / r23576;
        double r23578 = r23577 / r23574;
        return r23578;
}

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 18.5

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

  3. Applied times-frac1.4

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

  5. Applied *-un-lft-identity1.4

    \[\leadsto \frac{-\color{blue}{1 \cdot t1}}{t1 + u} \cdot \frac{v}{t1 + u}\]

  6. Applied distribute-lft-neg-in1.4

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

  7. Applied associate-/l*1.6

    \[\leadsto \color{blue}{\frac{-1}{\frac{t1 + u}{t1}}} \cdot \frac{v}{t1 + u}\]
  8. Using strategy rm

  9. Applied pow11.6

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

  10. Applied pow11.6

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

  11. Applied pow-prod-down1.6

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

  12. Simplified1.6

    \[\leadsto {\color{blue}{\left(\frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}\right)}}^{1}\]

  13. Final simplification1.6

    \[\leadsto \frac{\frac{-t1}{\frac{t1 + u}{v}}}{t1 + u}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))