Average Error: 1.0 → 0.0
Time: 3.0s
Precision: binary64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
↓
\[\frac{\frac{1.3333333333333333}{\pi - \pi \cdot {v}^{2}}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \]
\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{\frac{1.3333333333333333}{\pi - \pi \cdot {v}^{2}}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}}

(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 (- PI (* PI (pow v 2.0))))
  (sqrt (fma v (* v -6.0) 2.0))))
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 / (((double) M_PI) - (((double) M_PI) * pow(v, 2.0)))) / sqrt(fma(v, (v * -6.0), 2.0));
}

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. Simplified0.0

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

    Proof

  3. Taylor expanded in v around 0 0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022004 
(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)))))))