Average Error: 1.0 → 0.0
Time: 3.7s
Precision: binary64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}} \]
↓
\[-1.3333333333333333 \cdot \frac{\frac{\frac{1}{\pi}}{\mathsf{fma}\left(v, v, -1\right)}}{\sqrt{\mathsf{fma}\left(v, v \cdot -6, 2\right)}} \]
\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)}}
↓
-1.3333333333333333 \cdot \frac{\frac{\frac{1}{\pi}}{\mathsf{fma}\left(v, v, -1\right)}}{\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
  (/ (/ (/ 1.0 PI) (fma v v -1.0)) (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 * (((1.0 / ((double) M_PI)) / fma(v, v, -1.0)) / 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

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

  3. Applied *-un-lft-identity_binary640.0

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

  4. Applied sqrt-prod_binary640.0

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

  5. Applied div-inv_binary640.0

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

  6. Applied times-frac_binary640.0

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

  7. Simplified0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2022097 
(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)))))))