Average Error: 0.0 → 0.0
Time: 4.2s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt{2} \cdot \frac{\left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{\left(1 + v \cdot v\right) \cdot 4}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{2} \cdot \frac{\left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}{\left(1 + v \cdot v\right) \cdot 4}
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) * ((((1.0 * 1.0) - ((v * v) * (v * v))) * sqrt((1.0 - (3.0 * (v * v))))) / ((1.0 + (v * v)) * 4.0)));
}

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 div-inv0.0

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

  4. Applied associate-*l*0.0

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

  5. Applied associate-*l*0.0

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

  6. Simplified0.0

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

  8. Applied flip--0.0

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

  9. Applied frac-times0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020078 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))