Average Error: 0.5 → 0.5
Time: 11.3s
Precision: binary64
\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
↓
\[x1 + \left(x1 + \left(\left(x1 \cdot x1 + 1\right) \cdot \left(\frac{x1 \cdot \left(x1 \cdot 3\right) + \left(2 \cdot x2 - x1\right)}{x1 \cdot x1 + 1} \cdot \left(x1 \cdot \left(2 \cdot \left(\frac{x1 \cdot \left(x1 \cdot 3\right) + \left(2 \cdot x2 - x1\right)}{x1 \cdot x1 + 1} - 3\right)\right) + \left(x1 \cdot x1\right) \cdot 4\right) - x1 \cdot \left(x1 \cdot 6\right)\right) + \left(x1 \cdot \left(x1 \cdot \left(x1 + 3 \cdot \frac{x1 \cdot \left(x1 \cdot 3\right) + \left(2 \cdot x2 - x1\right)}{x1 \cdot x1 + 1}\right)\right) + 3 \cdot \frac{x1 \cdot \left(x1 \cdot 3\right) - \left(x1 + 2 \cdot x2\right)}{x1 \cdot x1 + 1}\right)\right)\right)\]
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)
↓
x1 + \left(x1 + \left(\left(x1 \cdot x1 + 1\right) \cdot \left(\frac{x1 \cdot \left(x1 \cdot 3\right) + \left(2 \cdot x2 - x1\right)}{x1 \cdot x1 + 1} \cdot \left(x1 \cdot \left(2 \cdot \left(\frac{x1 \cdot \left(x1 \cdot 3\right) + \left(2 \cdot x2 - x1\right)}{x1 \cdot x1 + 1} - 3\right)\right) + \left(x1 \cdot x1\right) \cdot 4\right) - x1 \cdot \left(x1 \cdot 6\right)\right) + \left(x1 \cdot \left(x1 \cdot \left(x1 + 3 \cdot \frac{x1 \cdot \left(x1 \cdot 3\right) + \left(2 \cdot x2 - x1\right)}{x1 \cdot x1 + 1}\right)\right) + 3 \cdot \frac{x1 \cdot \left(x1 \cdot 3\right) - \left(x1 + 2 \cdot x2\right)}{x1 \cdot x1 + 1}\right)\right)\right)
(FPCore (x1 x2)
 :precision binary64
 (+
  x1
  (+
   (+
    (+
     (+
      (*
       (+
        (*
         (*
          (* 2.0 x1)
          (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))
         (- (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)) 3.0))
        (*
         (* x1 x1)
         (-
          (* 4.0 (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))
          6.0)))
       (+ (* x1 x1) 1.0))
      (*
       (* (* 3.0 x1) x1)
       (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))
     (* (* x1 x1) x1))
    x1)
   (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))))
↓
(FPCore (x1 x2)
 :precision binary64
 (+
  x1
  (+
   x1
   (+
    (*
     (+ (* x1 x1) 1.0)
     (-
      (*
       (/ (+ (* x1 (* x1 3.0)) (- (* 2.0 x2) x1)) (+ (* x1 x1) 1.0))
       (+
        (*
         x1
         (*
          2.0
          (-
           (/ (+ (* x1 (* x1 3.0)) (- (* 2.0 x2) x1)) (+ (* x1 x1) 1.0))
           3.0)))
        (* (* x1 x1) 4.0)))
      (* x1 (* x1 6.0))))
    (+
     (*
      x1
      (*
       x1
       (+
        x1
        (*
         3.0
         (/ (+ (* x1 (* x1 3.0)) (- (* 2.0 x2) x1)) (+ (* x1 x1) 1.0))))))
     (* 3.0 (/ (- (* x1 (* x1 3.0)) (+ x1 (* 2.0 x2))) (+ (* x1 x1) 1.0))))))))
double code(double x1, double x2) {
	return ((double) (x1 + ((double) (((double) (((double) (((double) (((double) (((double) (((double) (((double) (((double) (2.0 * x1)) * (((double) (((double) (((double) (((double) (3.0 * x1)) * x1)) + ((double) (2.0 * x2)))) - x1)) / ((double) (((double) (x1 * x1)) + 1.0))))) * ((double) ((((double) (((double) (((double) (((double) (3.0 * x1)) * x1)) + ((double) (2.0 * x2)))) - x1)) / ((double) (((double) (x1 * x1)) + 1.0))) - 3.0)))) + ((double) (((double) (x1 * x1)) * ((double) (((double) (4.0 * (((double) (((double) (((double) (((double) (3.0 * x1)) * x1)) + ((double) (2.0 * x2)))) - x1)) / ((double) (((double) (x1 * x1)) + 1.0))))) - 6.0)))))) * ((double) (((double) (x1 * x1)) + 1.0)))) + ((double) (((double) (((double) (3.0 * x1)) * x1)) * (((double) (((double) (((double) (((double) (3.0 * x1)) * x1)) + ((double) (2.0 * x2)))) - x1)) / ((double) (((double) (x1 * x1)) + 1.0))))))) + ((double) (((double) (x1 * x1)) * x1)))) + x1)) + ((double) (3.0 * (((double) (((double) (((double) (((double) (3.0 * x1)) * x1)) - ((double) (2.0 * x2)))) - x1)) / ((double) (((double) (x1 * x1)) + 1.0)))))))));
}
↓
double code(double x1, double x2) {
	return ((double) (x1 + ((double) (x1 + ((double) (((double) (((double) (((double) (x1 * x1)) + 1.0)) * ((double) (((double) ((((double) (((double) (x1 * ((double) (x1 * 3.0)))) + ((double) (((double) (2.0 * x2)) - x1)))) / ((double) (((double) (x1 * x1)) + 1.0))) * ((double) (((double) (x1 * ((double) (2.0 * ((double) ((((double) (((double) (x1 * ((double) (x1 * 3.0)))) + ((double) (((double) (2.0 * x2)) - x1)))) / ((double) (((double) (x1 * x1)) + 1.0))) - 3.0)))))) + ((double) (((double) (x1 * x1)) * 4.0)))))) - ((double) (x1 * ((double) (x1 * 6.0)))))))) + ((double) (((double) (x1 * ((double) (x1 * ((double) (x1 + ((double) (3.0 * (((double) (((double) (x1 * ((double) (x1 * 3.0)))) + ((double) (((double) (2.0 * x2)) - x1)))) / ((double) (((double) (x1 * x1)) + 1.0))))))))))) + ((double) (3.0 * (((double) (((double) (x1 * ((double) (x1 * 3.0)))) - ((double) (x1 + ((double) (2.0 * x2)))))) / ((double) (((double) (x1 * x1)) + 1.0)))))))))))));
}

Error

Bits error versus x1

Bits error versus x2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

    \[\leadsto \color{blue}{x1 + \left(x1 + \left(\left(x1 \cdot x1 + 1\right) \cdot \left(\frac{x1 \cdot \left(x1 \cdot 3\right) + \left(2 \cdot x2 - x1\right)}{x1 \cdot x1 + 1} \cdot \left(x1 \cdot \left(2 \cdot \left(\frac{x1 \cdot \left(x1 \cdot 3\right) + \left(2 \cdot x2 - x1\right)}{x1 \cdot x1 + 1} - 3\right)\right) + 4 \cdot \left(x1 \cdot x1\right)\right) - 6 \cdot \left(x1 \cdot x1\right)\right) + \left(x1 \cdot \left(x1 \cdot \left(3 \cdot \frac{x1 \cdot \left(x1 \cdot 3\right) + \left(2 \cdot x2 - x1\right)}{x1 \cdot x1 + 1} + x1\right)\right) + 3 \cdot \frac{x1 \cdot \left(x1 \cdot 3\right) - \left(x1 + 2 \cdot x2\right)}{x1 \cdot x1 + 1}\right)\right)\right)}\]
  3. Using strategy rm

  4. Applied associate-*r*0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2020198 
(FPCore (x1 x2)
  :name "Rosa's FloatVsDoubleBenchmark"
  :precision binary64
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2.0 x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) (- (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)) 3.0)) (* (* x1 x1) (- (* 4.0 (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))) 6.0))) (+ (* x1 x1) 1.0)) (* (* (* 3.0 x1) x1) (/ (- (+ (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))) (* (* x1 x1) x1)) x1) (* 3.0 (/ (- (- (* (* 3.0 x1) x1) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))))))