Average Error: 1.0 → 0.0
Time: 1.9s
Precision: binary64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{1}{\pi} \cdot \frac{1.3333333333333333}{\left(1 - v \cdot v\right) \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}}\]
\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{1.3333333333333333}{\left(1 - v \cdot v\right) \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}}
(FPCore (v)
 :precision binary64
 (/ 4.0 (* (* (* 3.0 PI) (- 1.0 (* v v))) (sqrt (- 2.0 (* 6.0 (* v v)))))))
↓
(FPCore (v)
 :precision binary64
 (*
  (/ 1.0 PI)
  (/ 1.3333333333333333 (* (- 1.0 (* v v)) (sqrt (- 2.0 (* (* v v) 6.0)))))))
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)) * (1.3333333333333333 / ((double) (((double) (1.0 - ((double) (v * v)))) * ((double) sqrt(((double) (2.0 - ((double) (((double) (v * v)) * 6.0)))))))))));
}

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. Simplified0.0

    \[\leadsto \color{blue}{\frac{1.3333333333333333}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)}}\]
  3. Using strategy rm

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

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

  5. Applied times-frac_binary640.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020210 
(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)))))))