Average Error: 18.3 → 1.3
Time: 3.0s
Precision: 64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[-\frac{\frac{t1}{t1 + u} \cdot v}{t1 + u}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
-\frac{\frac{t1}{t1 + u} \cdot v}{t1 + u}
double f(double u, double v, double t1) {
        double r16633 = t1;
        double r16634 = -r16633;
        double r16635 = v;
        double r16636 = r16634 * r16635;
        double r16637 = u;
        double r16638 = r16633 + r16637;
        double r16639 = r16638 * r16638;
        double r16640 = r16636 / r16639;
        return r16640;
}
↓
double f(double u, double v, double t1) {
        double r16641 = t1;
        double r16642 = u;
        double r16643 = r16641 + r16642;
        double r16644 = r16641 / r16643;
        double r16645 = v;
        double r16646 = r16644 * r16645;
        double r16647 = r16646 / r16643;
        double r16648 = -r16647;
        return r16648;
}

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 18.3

    \[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
  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 neg-sub01.4

    \[\leadsto \frac{\color{blue}{0 - t1}}{t1 + u} \cdot \frac{v}{t1 + u}\]

  6. Applied div-sub1.4

    \[\leadsto \color{blue}{\left(\frac{0}{t1 + u} - \frac{t1}{t1 + u}\right)} \cdot \frac{v}{t1 + u}\]

  7. Simplified1.4

    \[\leadsto \left(\color{blue}{0} - \frac{t1}{t1 + u}\right) \cdot \frac{v}{t1 + u}\]
  8. Using strategy rm

  9. Applied div-inv1.5

    \[\leadsto \left(0 - \color{blue}{t1 \cdot \frac{1}{t1 + u}}\right) \cdot \frac{v}{t1 + u}\]
  10. Using strategy rm

  11. Applied sub0-neg1.5

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

  12. Applied distribute-lft-neg-out1.5

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

  13. Simplified1.3

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

  14. Final simplification1.3

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))