Average Error: 0.5 → 0.5
Time: 12.6s
Precision: binary64
\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
↓
\[x1 + \left(\left(x1 \cdot x1 + 1\right) \cdot \left(\frac{1}{\sqrt{x1 \cdot x1 + 1}} \cdot \left(\left(x1 \cdot \left(2 \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + x1 \cdot 4\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{\sqrt{x1 \cdot x1 + 1}}\right) - \left(x1 \cdot x1\right) \cdot 6\right) + \left(\left(x1 \cdot x1\right) \cdot \left(x1 + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\right)\right)\]
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(\left(x1 \cdot x1 + 1\right) \cdot \left(\frac{1}{\sqrt{x1 \cdot x1 + 1}} \cdot \left(\left(x1 \cdot \left(2 \cdot \left(\frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + x1 \cdot 4\right)\right) \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{\sqrt{x1 \cdot x1 + 1}}\right) - \left(x1 \cdot x1\right) \cdot 6\right) + \left(\left(x1 \cdot x1\right) \cdot \left(x1 + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 + 3 \cdot \frac{\left(x1 \cdot \left(x1 \cdot 3\right) - 2 \cdot x2\right) - x1}{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) 1.0)
    (-
     (*
      (/ 1.0 (sqrt (+ (* x1 x1) 1.0)))
      (*
       (*
        x1
        (+
         (*
          2.0
          (-
           (/ (- (+ (* x1 (* x1 3.0)) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0))
           3.0))
         (* x1 4.0)))
       (/ (- (+ (* x1 (* x1 3.0)) (* 2.0 x2)) x1) (sqrt (+ (* x1 x1) 1.0)))))
     (* (* x1 x1) 6.0)))
   (+
    (*
     (* x1 x1)
     (+
      x1
      (* 3.0 (/ (- (+ (* x1 (* x1 3.0)) (* 2.0 x2)) x1) (+ (* x1 x1) 1.0)))))
    (+
     x1
     (* 3.0 (/ (- (- (* x1 (* x1 3.0)) (* 2.0 x2)) x1) (+ (* 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) (((double) (((double) (((double) (x1 * x1)) + 1.0)) * ((double) (((double) ((1.0 / ((double) sqrt(((double) (((double) (x1 * x1)) + 1.0))))) * ((double) (((double) (x1 * ((double) (((double) (2.0 * ((double) ((((double) (((double) (((double) (x1 * ((double) (x1 * 3.0)))) + ((double) (2.0 * x2)))) - x1)) / ((double) (((double) (x1 * x1)) + 1.0))) - 3.0)))) + ((double) (x1 * 4.0)))))) * (((double) (((double) (((double) (x1 * ((double) (x1 * 3.0)))) + ((double) (2.0 * x2)))) - x1)) / ((double) sqrt(((double) (((double) (x1 * x1)) + 1.0))))))))) - ((double) (((double) (x1 * x1)) * 6.0)))))) + ((double) (((double) (((double) (x1 * x1)) * ((double) (x1 + ((double) (3.0 * (((double) (((double) (((double) (x1 * ((double) (x1 * 3.0)))) + ((double) (2.0 * x2)))) - x1)) / ((double) (((double) (x1 * x1)) + 1.0))))))))) + ((double) (x1 + ((double) (3.0 * (((double) (((double) (((double) (x1 * ((double) (x1 * 3.0)))) - ((double) (2.0 * x2)))) - x1)) / ((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

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

  4. Applied add-sqr-sqrt_binary640.5

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

  5. Applied *-un-lft-identity_binary640.5

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

  6. Applied times-frac_binary640.5

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

  7. Applied associate-*l*_binary640.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2020205 
(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))))))