Average Error: 12.8 → 0.3
Time: 8.4s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left|w \cdot r\right|}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{\left|w \cdot r\right|}}\right) - 4.5\]
\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
↓
\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left|w \cdot r\right|}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{\left|w \cdot r\right|}}\right) - 4.5
double code(double v, double w, double r) {
	return (((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5);
}
↓
double code(double v, double w, double r) {
	return (((3.0 + (2.0 / (r * r))) - (fabs((w * r)) / (((1.0 - v) / (0.375 - (0.25 * v))) / fabs((w * r))))) - 4.5);
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.8

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

  3. Applied add-sqr-sqrt12.8

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

  4. Simplified12.8

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

  5. Simplified6.7

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

  7. Applied clear-num6.7

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

  8. Simplified0.4

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

  9. Taylor expanded around 0 0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{\frac{\frac{1 - v}{\color{blue}{0.375 - 0.25 \cdot v}}}{{\left(\left|w \cdot r\right|\right)}^{2}}}\right) - 4.5\]
  10. Using strategy rm

  11. Applied unpow20.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{\color{blue}{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}}}\right) - 4.5\]

  12. Applied *-un-lft-identity0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{\frac{\frac{1 - v}{\color{blue}{1 \cdot \left(0.375 - 0.25 \cdot v\right)}}}{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}}\right) - 4.5\]

  13. Applied *-un-lft-identity0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{\frac{\frac{\color{blue}{1 \cdot \left(1 - v\right)}}{1 \cdot \left(0.375 - 0.25 \cdot v\right)}}{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}}\right) - 4.5\]

  14. Applied times-frac0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{\frac{\color{blue}{\frac{1}{1} \cdot \frac{1 - v}{0.375 - 0.25 \cdot v}}}{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}}\right) - 4.5\]

  15. Applied times-frac0.4

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{1}{\color{blue}{\frac{\frac{1}{1}}{\left|w \cdot r\right|} \cdot \frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{\left|w \cdot r\right|}}}\right) - 4.5\]

  16. Applied associate-/r*0.3

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\frac{1}{\frac{\frac{1}{1}}{\left|w \cdot r\right|}}}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{\left|w \cdot r\right|}}}\right) - 4.5\]

  17. Simplified0.3

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left|w \cdot r\right|}}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{\left|w \cdot r\right|}}\right) - 4.5\]

  18. Final simplification0.3

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left|w \cdot r\right|}{\frac{\frac{1 - v}{0.375 - 0.25 \cdot v}}{\left|w \cdot r\right|}}\right) - 4.5\]

Reproduce

herbie shell --seed 2020078 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))