Average Error: 12.2 → 3.1
Time: 3.6m
Precision: 64
Internal Precision: 384
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\begin{array}{l} \mathbf{if}\;r \le -1.8777416836098266 \cdot 10^{+181}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)}{1 - v}\right) - 4.5\\ \mathbf{if}\;r \le -1.2556715329124771 \cdot 10^{-26}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}\right) - 4.5\\ \mathbf{if}\;r \le 1.7324129367663798 \cdot 10^{-59}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\sqrt[3]{\left(\frac{3 + -2 \cdot v}{1 - v \cdot v} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)} \cdot \sqrt[3]{\left(\frac{3 + -2 \cdot v}{1 - v \cdot v} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)}\right) \cdot \sqrt[3]{\left(\frac{3 + -2 \cdot v}{1 - v \cdot v} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)}\right) - 4.5\\ \mathbf{if}\;r \le 8.143179940192371 \cdot 10^{+159}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\left(\sqrt[3]{3 + v \cdot -2} \cdot \sqrt[3]{3 + v \cdot -2}\right) \cdot \frac{\sqrt[3]{3 + -2 \cdot v}}{1 - v \cdot v}\right) \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)\right) - 4.5\\ \end{array}\]

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Split input into 4 regimes

  2. if r < -1.8777416836098266e+181

    1. Initial program 30.4

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
    2. Using strategy rm

    3. Applied associate-*l*10.0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\color{blue}{\left(w \cdot \left(w \cdot r\right)\right)} \cdot r\right)}{1 - v}\right) - 4.5\]

    if -1.8777416836098266e+181 < r < -1.2556715329124771e-26 or 1.7324129367663798e-59 < r < 8.143179940192371e+159

    1. Initial program 6.4

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
    2. Using strategy rm

    3. Applied associate-/l*1.1

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}}\right) - 4.5\]

    if -1.2556715329124771e-26 < r < 1.7324129367663798e-59

    1. Initial program 11.7

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
    2. Using strategy rm

    3. Applied flip--11.7

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{\color{blue}{\frac{1 \cdot 1 - v \cdot v}{1 + v}}}\right) - 4.5\]

    4. Applied associate-/r/11.7

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 \cdot 1 - v \cdot v} \cdot \left(1 + v\right)}\right) - 4.5\]

    5. Applied simplify2.1

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{3 + -2 \cdot v}{1 - v \cdot v} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right)} \cdot \left(1 + v\right)\right) - 4.5\]
    6. Using strategy rm

    7. Applied add-cube-cbrt2.1

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\sqrt[3]{\left(\frac{3 + -2 \cdot v}{1 - v \cdot v} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)} \cdot \sqrt[3]{\left(\frac{3 + -2 \cdot v}{1 - v \cdot v} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)}\right) \cdot \sqrt[3]{\left(\frac{3 + -2 \cdot v}{1 - v \cdot v} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)}}\right) - 4.5\]

    if 8.143179940192371e+159 < r

    1. Initial program 27.6

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
    2. Using strategy rm

    3. Applied flip--28.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{\color{blue}{\frac{1 \cdot 1 - v \cdot v}{1 + v}}}\right) - 4.5\]

    4. Applied associate-/r/28.6

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 \cdot 1 - v \cdot v} \cdot \left(1 + v\right)}\right) - 4.5\]

    5. Applied simplify11.0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{3 + -2 \cdot v}{1 - v \cdot v} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right)} \cdot \left(1 + v\right)\right) - 4.5\]
    6. Using strategy rm

    7. Applied *-un-lft-identity11.0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{3 + -2 \cdot v}{\color{blue}{1 \cdot \left(1 - v \cdot v\right)}} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)\right) - 4.5\]

    8. Applied add-cube-cbrt11.1

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{\left(\sqrt[3]{3 + -2 \cdot v} \cdot \sqrt[3]{3 + -2 \cdot v}\right) \cdot \sqrt[3]{3 + -2 \cdot v}}}{1 \cdot \left(1 - v \cdot v\right)} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)\right) - 4.5\]

    9. Applied times-frac11.2

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{\sqrt[3]{3 + -2 \cdot v} \cdot \sqrt[3]{3 + -2 \cdot v}}{1} \cdot \frac{\sqrt[3]{3 + -2 \cdot v}}{1 - v \cdot v}\right)} \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)\right) - 4.5\]

    10. Applied simplify11.2

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\color{blue}{\left(\sqrt[3]{3 + v \cdot -2} \cdot \sqrt[3]{3 + v \cdot -2}\right)} \cdot \frac{\sqrt[3]{3 + -2 \cdot v}}{1 - v \cdot v}\right) \cdot \left(0.125 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\right) \cdot \left(1 + v\right)\right) - 4.5\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))