Average Error: 0.4 → 0.5
Time: 12.1s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(3 \cdot \frac{t \cdot \left({v}^{4} \cdot \pi\right)}{\sqrt{2}} + t \cdot \left(\pi \cdot \sqrt{2}\right)\right) - \left(4.5 \cdot \frac{t \cdot \left({v}^{4} \cdot \pi\right)}{{\left(\sqrt{2}\right)}^{3}} + \left(3 \cdot \frac{t \cdot \left(\pi \cdot {v}^{2}\right)}{\sqrt{2}} + t \cdot \left(\left(\pi \cdot \sqrt{2}\right) \cdot {v}^{2}\right)\right)\right)}\]
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}
↓
\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(3 \cdot \frac{t \cdot \left({v}^{4} \cdot \pi\right)}{\sqrt{2}} + t \cdot \left(\pi \cdot \sqrt{2}\right)\right) - \left(4.5 \cdot \frac{t \cdot \left({v}^{4} \cdot \pi\right)}{{\left(\sqrt{2}\right)}^{3}} + \left(3 \cdot \frac{t \cdot \left(\pi \cdot {v}^{2}\right)}{\sqrt{2}} + t \cdot \left(\left(\pi \cdot \sqrt{2}\right) \cdot {v}^{2}\right)\right)\right)}
(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))
↓
(FPCore (v t)
 :precision binary64
 (/
  (- 1.0 (* 5.0 (* v v)))
  (-
   (+ (* 3.0 (/ (* t (* (pow v 4.0) PI)) (sqrt 2.0))) (* t (* PI (sqrt 2.0))))
   (+
    (* 4.5 (/ (* t (* (pow v 4.0) PI)) (pow (sqrt 2.0) 3.0)))
    (+
     (* 3.0 (/ (* t (* PI (pow v 2.0))) (sqrt 2.0)))
     (* t (* (* PI (sqrt 2.0)) (pow v 2.0))))))))
double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((((double) M_PI) * t) * sqrt(2.0 * (1.0 - (3.0 * (v * v))))) * (1.0 - (v * v)));
}
↓
double code(double v, double t) {
	return (1.0 - (5.0 * (v * v))) / (((3.0 * ((t * (pow(v, 4.0) * ((double) M_PI))) / sqrt(2.0))) + (t * (((double) M_PI) * sqrt(2.0)))) - ((4.5 * ((t * (pow(v, 4.0) * ((double) M_PI))) / pow(sqrt(2.0), 3.0))) + ((3.0 * ((t * (((double) M_PI) * pow(v, 2.0))) / sqrt(2.0))) + (t * ((((double) M_PI) * sqrt(2.0)) * pow(v, 2.0))))));
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

  2. Simplified0.4

    \[\leadsto \color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 + v \cdot \left(-6 \cdot v\right)} \cdot \left(\left(\pi \cdot t\right) \cdot \left(1 - v \cdot v\right)\right)}}\]

  3. Taylor expanded around 0 0.5

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(3 \cdot \frac{t \cdot \left({v}^{4} \cdot \pi\right)}{\sqrt{2}} + t \cdot \left(\pi \cdot \sqrt{2}\right)\right) - \left(4.5 \cdot \frac{t \cdot \left({v}^{4} \cdot \pi\right)}{{\left(\sqrt{2}\right)}^{3}} + \left(3 \cdot \frac{t \cdot \left({v}^{2} \cdot \pi\right)}{\sqrt{2}} + t \cdot \left({v}^{2} \cdot \left(\pi \cdot \sqrt{2}\right)\right)\right)\right)}}\]

  4. Final simplification0.5

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(3 \cdot \frac{t \cdot \left({v}^{4} \cdot \pi\right)}{\sqrt{2}} + t \cdot \left(\pi \cdot \sqrt{2}\right)\right) - \left(4.5 \cdot \frac{t \cdot \left({v}^{4} \cdot \pi\right)}{{\left(\sqrt{2}\right)}^{3}} + \left(3 \cdot \frac{t \cdot \left(\pi \cdot {v}^{2}\right)}{\sqrt{2}} + t \cdot \left(\left(\pi \cdot \sqrt{2}\right) \cdot {v}^{2}\right)\right)\right)}\]

Reproduce

herbie shell --seed 2021075 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))