Average Error: 0.6 → 0.1
Time: 3.7m
Precision: 64
Internal precision: 1408
\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
⬇
\[\left(\left(x1 + 9\right) \cdot \left(x1 \cdot x1\right) - \left(x2 \cdot 6 + x1\right)\right) + \left(\left(\frac{\left(x2 + \left(x2 - x1\right)\right) + {x1}^2 \cdot 3}{\frac{1 + {x1}^2}{4}} - 6\right) \cdot \left({x1}^2 \cdot {x1}^2 + {x1}^2\right) + \left(\left(\left(1 + {x1}^2\right) \cdot \left(\left(x2 + \left(x2 - x1\right)\right) + {x1}^2 \cdot 3\right)\right) \cdot \frac{x1 + x1}{1 + {x1}^2}\right) \cdot \left(\frac{\left(x2 + \left(x2 - x1\right)\right) + {x1}^2 \cdot 3}{1 + {x1}^2} - 3\right)\right)\]

Error

Bits error versus x1

Bits error versus x2

Derivation

  1. Initial program 0.6

    \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]

  2. Applied simplify 0.6

    \[\leadsto \color{blue}{\left(\left(x1 + \left(\frac{\left(\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3\right) \cdot \left(\left(x1 \cdot x1\right) \cdot 3\right)}{1 + x1 \cdot x1} + \frac{3}{1 + x1 \cdot x1} \cdot \left(\left(x1 \cdot x1\right) \cdot 3 - \left(x1 + \left(x2 + x2\right)\right)\right)\right)\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\frac{\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3}{\frac{1 + x1 \cdot x1}{4}} - 6\right) \cdot \left(x1 \cdot x1\right) + \frac{x1 + x1}{\frac{1 + x1 \cdot x1}{\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3}} \cdot \left(\frac{\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3}{1 + x1 \cdot x1} - 3\right)\right)\right) + \left(x1 + {x1}^3\right)}\]

  3. Applied taylor 0.2

    \[\leadsto \left(\left(9 \cdot {x1}^2 - \left(2 \cdot x1 + 6 \cdot x2\right)\right) + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\frac{\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3}{\frac{1 + x1 \cdot x1}{4}} - 6\right) \cdot \left(x1 \cdot x1\right) + \frac{x1 + x1}{\frac{1 + x1 \cdot x1}{\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3}} \cdot \left(\frac{\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3}{1 + x1 \cdot x1} - 3\right)\right)\right) + \left(x1 + {x1}^3\right)\]

  4. Taylor expanded around 0 0.2

    \[\leadsto \left(\color{blue}{\left(9 \cdot {x1}^2 - \left(2 \cdot x1 + 6 \cdot x2\right)\right)} + \left(1 + x1 \cdot x1\right) \cdot \left(\left(\frac{\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3}{\frac{1 + x1 \cdot x1}{4}} - 6\right) \cdot \left(x1 \cdot x1\right) + \frac{x1 + x1}{\frac{1 + x1 \cdot x1}{\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3}} \cdot \left(\frac{\left(x2 + \left(x2 - x1\right)\right) + \left(x1 \cdot x1\right) \cdot 3}{1 + x1 \cdot x1} - 3\right)\right)\right) + \left(x1 + {x1}^3\right)\]

  5. Applied simplify 0.1

    \[\leadsto \color{blue}{\left(\left(\left({x1}^3 + x1\right) + x1 \cdot \left(x1 \cdot 9\right)\right) - \left(\left(x1 + x1\right) + x2 \cdot 6\right)\right) + \left(\left(\frac{\left(x2 + \left(x2 - x1\right)\right) + {x1}^2 \cdot 3}{\frac{1 + {x1}^2}{4}} - 6\right) \cdot \left({x1}^2 \cdot {x1}^2 + {x1}^2\right) + \left(\left(\left(1 + {x1}^2\right) \cdot \left(\left(x2 + \left(x2 - x1\right)\right) + {x1}^2 \cdot 3\right)\right) \cdot \frac{x1 + x1}{1 + {x1}^2}\right) \cdot \left(\frac{\left(x2 + \left(x2 - x1\right)\right) + {x1}^2 \cdot 3}{1 + {x1}^2} - 3\right)\right)}\]

  6. Applied simplify 0.1

    \[\leadsto \color{blue}{\left(\left(x1 + 9\right) \cdot \left(x1 \cdot x1\right) - \left(x2 \cdot 6 + x1\right)\right)} + \left(\left(\frac{\left(x2 + \left(x2 - x1\right)\right) + {x1}^2 \cdot 3}{\frac{1 + {x1}^2}{4}} - 6\right) \cdot \left({x1}^2 \cdot {x1}^2 + {x1}^2\right) + \left(\left(\left(1 + {x1}^2\right) \cdot \left(\left(x2 + \left(x2 - x1\right)\right) + {x1}^2 \cdot 3\right)\right) \cdot \frac{x1 + x1}{1 + {x1}^2}\right) \cdot \left(\frac{\left(x2 + \left(x2 - x1\right)\right) + {x1}^2 \cdot 3}{1 + {x1}^2} - 3\right)\right)\]
  7. Removed slow pow expressions

Runtime

Time bar (total: 3.7m) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (x1 x2)
  :name "jetEngine"
  :pre (and (<= -5 x1 5) (<= -20 x2 5))
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2 x1) (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))) (- (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1)) 3)) (* (* x1 x1) (- (* 4 (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))) 6))) (+ (* x1 x1) 1)) (* (* (* 3 x1) x1) (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1)))) (* (* x1 x1) x1)) x1) (* 3 (/ (- (- (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))))))