Average Error: 0.0 → 0.0
Time: 2.3s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right)}{4}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(\sqrt{2} \cdot \left(1 - v \cdot v\right)\right)}{4}
double code(double v) {
	return ((double) (((double) ((((double) sqrt(2.0)) / 4.0) * ((double) sqrt(((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))) * ((double) (1.0 - ((double) (v * v))))));
}
↓
double code(double v) {
	return (((double) (((double) sqrt(((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))) * ((double) (((double) sqrt(2.0)) * ((double) (1.0 - ((double) (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 Error: 0.0 bits

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

  2. SimplifiedError: 0.0 bits

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

  4. Applied associate-*r/Error: 0.0 bits

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

  5. Applied associate-*r/Error: 0.0 bits

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

  6. SimplifiedError: 0.0 bits

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

  7. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020200 
(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))))