Average Error: 1.0 → 0.0
Time: 3.2s
Precision: binary64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\pi} \cdot \frac{4}{\left(1 - v \cdot v\right) \cdot \left(\sqrt{2 - v \cdot \left(v \cdot 6\right)} \cdot 3\right)}\]
\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{1}{\pi} \cdot \frac{4}{\left(1 - v \cdot v\right) \cdot \left(\sqrt{2 - v \cdot \left(v \cdot 6\right)} \cdot 3\right)}
double code(double v) {
	return (4.0 / ((double) (((double) (((double) (3.0 * ((double) M_PI))) * ((double) (1.0 - ((double) (v * v)))))) * ((double) sqrt(((double) (2.0 - ((double) (6.0 * ((double) (v * v)))))))))));
}
↓
double code(double v) {
	return ((double) ((1.0 / ((double) M_PI)) * (4.0 / ((double) (((double) (1.0 - ((double) (v * v)))) * ((double) (((double) sqrt(((double) (2.0 - ((double) (v * ((double) (v * 6.0)))))))) * 3.0)))))));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified1.0

    \[\leadsto \color{blue}{\frac{4}{3 \cdot \left(\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - v \cdot \left(v \cdot 6\right)}\right)\right)}}\]
  3. Using strategy rm

  4. Applied associate-/r*0.0

    \[\leadsto \color{blue}{\frac{\frac{4}{3}}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - v \cdot \left(v \cdot 6\right)}\right)}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.0

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

  7. Applied *-un-lft-identity0.0

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

  8. Applied times-frac0.0

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

  9. Applied times-frac0.0

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

  10. Simplified0.0

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

  11. Simplified0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020182 
(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)))))))