Average Error: 18.1 → 1.0
Time: 4.0s
Precision: binary64
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;u \leq -7.086882964238897 \cdot 10^{+112} \lor \neg \left(u \leq 2.0160755146411715 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(u + t1\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \end{array}\]
\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}
↓
\begin{array}{l}
\mathbf{if}\;u \leq -7.086882964238897 \cdot 10^{+112} \lor \neg \left(u \leq 2.0160755146411715 \cdot 10^{+39}\right):\\
\;\;\;\;\frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{v}{\left(u + t1\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\

\end{array}
(FPCore (u v t1) :precision binary64 (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))
↓
(FPCore (u v t1)
 :precision binary64
 (if (or (<= u -7.086882964238897e+112) (not (<= u 2.0160755146411715e+39)))
   (/ (/ v (+ u t1)) (- -1.0 (/ u t1)))
   (/ v (* (+ u t1) (- -1.0 (/ u t1))))))
double code(double u, double v, double t1) {
	return (-t1 * v) / ((t1 + u) * (t1 + u));
}
↓
double code(double u, double v, double t1) {
	double tmp;
	if ((u <= -7.086882964238897e+112) || !(u <= 2.0160755146411715e+39)) {
		tmp = (v / (u + t1)) / (-1.0 - (u / t1));
	} else {
		tmp = v / ((u + t1) * (-1.0 - (u / t1)));
	}
	return tmp;
}

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. Split input into 2 regimes

  2. if u < -7.086882964238897e112 or 2.01607551464117149e39 < u

    1. Initial program 15.8

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

    2. Simplified1.3

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

    if -7.086882964238897e112 < u < 2.01607551464117149e39

    1. Initial program 19.6

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

    2. Simplified1.6

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

    4. Applied associate-/l/_binary640.8

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

    5. Simplified0.8

      \[\leadsto \frac{v}{\color{blue}{\left(t1 + u\right) \cdot \left(-1 - \frac{u}{t1}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;u \leq -7.086882964238897 \cdot 10^{+112} \lor \neg \left(u \leq 2.0160755146411715 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{\frac{v}{u + t1}}{-1 - \frac{u}{t1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{v}{\left(u + t1\right) \cdot \left(-1 - \frac{u}{t1}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  :precision binary64
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))