Falkner and Boettcher, Appendix B, 2

Average Error: 0.0 → 0.0

Time: 1.5s

Precision: binary64

Cost: 13888

Math

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

↓

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

FPCore

(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 (* v v)) (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* (* v v) 3.0))))))

C

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 - (v * v)) * ((sqrt(2.0) / 4.0) * sqrt(1.0 - ((v * v) * 3.0)));
}

Error

Bits error versus v

Alternatives

Alternative 1
Error	0.2
Cost	13696

\[\sqrt{2} \cdot \left(\left(0.25 + {v}^{4} \cdot 0.09375\right) - \left(v \cdot v\right) \cdot 0.625\right)\]

Alternative 2
Error	0.3
Cost	7360

\[\left(1 - v \cdot v\right) \cdot \left(\sqrt{2} \cdot \left(0.25 + \left(v \cdot v\right) \cdot -0.375\right)\right)\]

Alternative 3
Error	0.7
Cost	6592

\[\sqrt{2} \cdot 0.25\]

Alternative 4
Error	52.7
Cost	64

\[1\]

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. Simplified 0.0

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

3. Final simplification 0.0

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

Reproduce

herbie shell --seed 2021044 
(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))))