Average Error: 0.4 → 0.3
Time: 6.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{t \cdot \left(\sqrt{2 - 6 \cdot {v}^{2}} \cdot \pi - \sqrt{2 - 6 \cdot {v}^{2}} \cdot \left({v}^{2} \cdot \pi\right)\right)}\]
\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)}{t \cdot \left(\sqrt{2 - 6 \cdot {v}^{2}} \cdot \pi - \sqrt{2 - 6 \cdot {v}^{2}} \cdot \left({v}^{2} \cdot \pi\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)))
  (*
   t
   (-
    (* (sqrt (- 2.0 (* 6.0 (pow v 2.0)))) PI)
    (* (sqrt (- 2.0 (* 6.0 (pow v 2.0)))) (* (pow v 2.0) PI))))))
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))) / (t * ((sqrt(2.0 - (6.0 * pow(v, 2.0))) * ((double) M_PI)) - (sqrt(2.0 - (6.0 * pow(v, 2.0))) * (pow(v, 2.0) * ((double) M_PI)))));
}

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 0.3

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

  4. Final simplification0.3

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

Alternatives

Reproduce

herbie shell --seed 2021113 
(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)))))