Falkner and Boettcher, Equation (22+)

Average Error: 1.0 → 0.0

Time: 2.1s

Precision: binary64

Math

\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

↓

\[\frac{-1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\pi \cdot \mathsf{fma}\left(v, v, -1\right)\right)} \]

FPCore

(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))

↓

(FPCore (v)
 :precision binary64
 (/
  -1.3333333333333333
  (* (sqrt (fma v (* v -6.0) 2.0)) (* PI (fma v v -1.0)))))

C

double code(double v) {
	return 4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt(2.0 - (6.0 * (v * v))));
}

↓

double code(double v) {
	return -1.3333333333333333 / (sqrt(fma(v, (v * -6.0), 2.0)) * (((double) M_PI) * fma(v, v, -1.0)));
}

Error

Bits error versus v

Derivation

1. Initial program 1.0

   \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{\frac{-1.3333333333333333}{\pi \cdot \mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}} \]

3. Applied associate-/l/_binary64 0.0

   \[\leadsto \color{blue}{\frac{-1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\pi \cdot \mathsf{fma}\left(v, v, -1\right)\right)}} \]

4. Final simplification 0.0

   \[\leadsto \frac{-1.3333333333333333}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)} \cdot \left(\pi \cdot \mathsf{fma}\left(v, v, -1\right)\right)} \]

Reproduce

herbie shell --seed 2022077 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))