Average Error: 0.0 → 0.0
Time: 2.5s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(1 - {v}^{6}\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + {v}^{4} \cdot -9}\right)}{\left({v}^{4} + \left(1 + v \cdot v\right)\right) \cdot \left(4 \cdot \sqrt{1 + \left(v \cdot v\right) \cdot 3}\right)}\]
\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(1 - {v}^{6}\right) \cdot \left(\sqrt{2} \cdot \sqrt{1 + {v}^{4} \cdot -9}\right)}{\left({v}^{4} + \left(1 + v \cdot v\right)\right) \cdot \left(4 \cdot \sqrt{1 + \left(v \cdot v\right) \cdot 3}\right)}
(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v)
 :precision binary64
 (/
  (* (- 1.0 (pow v 6.0)) (* (sqrt 2.0) (sqrt (+ 1.0 (* (pow v 4.0) -9.0)))))
  (* (+ (pow v 4.0) (+ 1.0 (* v v))) (* 4.0 (sqrt (+ 1.0 (* (* v v) 3.0)))))))
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 ((1.0 - pow(v, 6.0)) * (sqrt(2.0) * sqrt(1.0 + (pow(v, 4.0) * -9.0)))) / ((pow(v, 4.0) + (1.0 + (v * v))) * (4.0 * sqrt(1.0 + ((v * v) * 3.0))));
}

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 flip3--_binary64_14460.0

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

  4. Applied flip--_binary64_14170.0

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

  5. Applied sqrt-div_binary64_14590.0

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

  6. Applied frac-times_binary64_14520.0

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

  7. Applied frac-times_binary64_14520.0

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

  8. Simplified0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020342 
(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))))