Average Error: 0.0 → 0.0
Time: 1.4min
Precision: binary64
Cost: 47424
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(4 \cdot \sqrt{3 \cdot \left(v \cdot v\right) + 1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}} - \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}}\right)\right)}{\left(3 \cdot \left(v \cdot v\right) + 1\right) \cdot 16}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(4 \cdot \sqrt{3 \cdot \left(v \cdot v\right) + 1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}} - \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}}\right)\right)}{\left(3 \cdot \left(v \cdot v\right) + 1\right) \cdot 16}
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (/
  (*
   (* 4.0 (sqrt (+ (* 3.0 (* v v)) 1.0)))
   (-
    (* (sqrt 2.0) (sqrt (+ 1.0 (* -9.0 (pow v 4.0)))))
    (* (* v v) (* (sqrt 2.0) (sqrt (+ 1.0 (* -9.0 (pow v 4.0))))))))
  (* (+ (* 3.0 (* v v)) 1.0) 16.0)))
double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt(1.0 - (3.0 * (v * v)))) * (1.0 - (v * v));
}
↓
double code(double v) {
	return ((4.0 * sqrt((3.0 * (v * v)) + 1.0)) * ((sqrt(2.0) * sqrt(1.0 + (-9.0 * pow(v, 4.0)))) - ((v * v) * (sqrt(2.0) * sqrt(1.0 + (-9.0 * pow(v, 4.0))))))) / (((3.0 * (v * v)) + 1.0) * 16.0);
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.0
Cost27328
\[\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)} - \left(v \cdot v\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\]
Alternative 2
Error0.0
Cost13888
\[\frac{\left(1 - v \cdot v\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)}{4}\]
Alternative 3
Error0.2
Cost13696
\[\sqrt{2} \cdot \left(\left(0.25 + {v}^{4} \cdot 0.09375\right) - \left(v \cdot v\right) \cdot 0.625\right)\]
Alternative 4
Error0.3
Cost6976
\[\sqrt{2} \cdot \left(0.25 - \left(v \cdot v\right) \cdot 0.625\right)\]
Alternative 5
Error0.7
Cost6592
\[\sqrt{2} \cdot 0.25\]
Alternative 6
Error52.7
Cost64
\[1\]

Error

Time

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  2. Using strategy rm

  3. Applied sub-neg_binary64_21170.0

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

  4. Applied distribute-rgt-in_binary64_20740.0

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

  5. Simplified0.0

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

  6. Simplified0.0

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

  8. Applied flip--_binary64_20990.0

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

  9. Applied sqrt-div_binary64_21410.0

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

  10. Applied frac-times_binary64_21340.0

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

  11. Applied associate-*l/_binary64_20670.0

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

  12. Applied flip--_binary64_20990.0

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

  13. Applied sqrt-div_binary64_21410.0

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

  14. Applied frac-times_binary64_21340.0

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

  15. Applied frac-add_binary64_21320.0

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

  16. Simplified0.0

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

  17. Simplified0.0

    \[\leadsto \frac{\left(4 \cdot \sqrt{3 \cdot \left(v \cdot v\right) + 1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}} - \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}}\right)\right)}{\color{blue}{16 \cdot \left(3 \cdot \left(v \cdot v\right) + 1\right)}}\]
  18. Using strategy rm

  19. Applied *-commutative_binary64_20550.0

    \[\leadsto \frac{\left(4 \cdot \sqrt{3 \cdot \left(v \cdot v\right) + 1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}} - \color{blue}{\left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}}\right) \cdot \left(v \cdot v\right)}\right)}{16 \cdot \left(3 \cdot \left(v \cdot v\right) + 1\right)}\]

  20. Simplified0.0

    \[\leadsto \color{blue}{\frac{\left(4 \cdot \sqrt{3 \cdot \left(v \cdot v\right) + 1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}} - \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}}\right) \cdot \left(v \cdot v\right)\right)}{16 \cdot \left(3 \cdot \left(v \cdot v\right) + 1\right)}}\]

  21. Final simplification0.0

    \[\leadsto \frac{\left(4 \cdot \sqrt{3 \cdot \left(v \cdot v\right) + 1}\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}} - \left(v \cdot v\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + -9 \cdot {v}^{4}}\right)\right)}{\left(3 \cdot \left(v \cdot v\right) + 1\right) \cdot 16}\]

Reproduce

herbie shell --seed 2021045 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))