Average Error: 17.5 → 1.0
Time: 30.7s
Precision: 64
Internal Precision: 384
\[\frac{\left(-t1\right) \cdot v}{\left(t1 + u\right) \cdot \left(t1 + u\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u} \le -7.159429525164284 \cdot 10^{-244}:\\ \;\;\;\;\left(\frac{-t1}{t1 + u} \cdot v\right) \cdot \frac{1}{t1 + u}\\ \mathbf{if}\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u} \le -0.0:\\ \;\;\;\;\frac{\frac{\left(-t1\right) \cdot v}{t1 + u}}{t1 + u}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}\\ \end{array}\]

Error

Bits error versus u

Bits error versus v

Bits error versus t1

Derivation

  1. Split input into 3 regimes

  2. if (* (/ (- t1) (+ t1 u)) (/ v (+ t1 u))) < -7.159429525164284e-244

    1. Initial program 27.0

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

    3. Applied times-frac1.5

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

    5. Applied div-inv1.7

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

    6. Applied associate-*r*0.8

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

    if -7.159429525164284e-244 < (* (/ (- t1) (+ t1 u)) (/ v (+ t1 u))) < -0.0

    1. Initial program 3.5

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

    3. Applied associate-/r*1.0

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

    if -0.0 < (* (/ (- t1) (+ t1 u)) (/ v (+ t1 u)))

    1. Initial program 26.8

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

    3. Applied times-frac1.3

      \[\leadsto \color{blue}{\frac{-t1}{t1 + u} \cdot \frac{v}{t1 + u}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 30.7s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(FPCore (u v t1)
  :name "Rosa's DopplerBench"
  (/ (* (- t1) v) (* (+ t1 u) (+ t1 u))))