Average Error: 0.0 → 0.0
Time: 2.8s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\sqrt{0.125 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right) \]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{0.125 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\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
 (* (sqrt (* 0.125 (- 1.0 (* 3.0 (* v v))))) (- 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(0.125 * (1.0 - (3.0 * (v * v)))) * (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. Applied add-sqr-sqrt_binary641.0

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

  3. Applied sqrt-unprod_binary640.0

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

  4. Simplified0.0

    \[\leadsto \sqrt{\color{blue}{0.125 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \left(1 - v \cdot v\right) \]

  5. Final simplification0.0

    \[\leadsto \sqrt{0.125 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right) \]

Reproduce

herbie shell --seed 2021215 
(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))))