Average Error: 0.0 → 0.0
Time: 3.6s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\frac{\frac{\left(\sqrt{2} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\right) \cdot \left(1 - {v}^{4}\right)}{4}}{1 + v \cdot v} \]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\frac{\left(\sqrt{2} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}\right) \cdot \left(1 - {v}^{4}\right)}{4}}{1 + v \cdot v}

(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (/
  (/ (* (* (sqrt 2.0) (sqrt (- 1.0 (* (* v v) 3.0)))) (- 1.0 (pow v 4.0))) 4.0)
  (+ 1.0 (* v v))))
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 (((sqrt(2.0) * sqrt(1.0 - ((v * v) * 3.0))) * (1.0 - pow(v, 4.0))) / 4.0) / (1.0 + (v * v));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 add-sqr-sqrt_binary640.0

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

  4. Applied add-sqr-sqrt_binary640.0

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

  5. Applied add-sqr-sqrt_binary641.0

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

  6. Applied times-frac_binary641.0

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

  7. Applied unswap-sqr_binary641.0

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

  8. Simplified1.0

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

  9. Simplified1.0

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

  11. Applied flip--_binary641.0

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

  12. Applied associate-*r/_binary641.0

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2021198 
(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))))