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

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

  3. Applied associate-/r*Error: 0.0 bits

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

  4. SimplifiedError: 0.0 bits

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

  5. Final simplificationError: 0.0 bits

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

Reproduce

herbie shell --seed 2020204 
(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)))))))