Average Error: 0.4 → 0.3
Time: 8.1s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(t \cdot \left(\pi \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}}}{1 - v \cdot v} \]
\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{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(t \cdot \left(\pi \cdot \sqrt{2}\right)\right) \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 3}}}{1 - v \cdot v}

(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 (* PI (sqrt 2.0))) (sqrt (- 1.0 (* (* v v) 3.0)))))
  (- 1.0 (* v v))))
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 * (((double) M_PI) * sqrt(2.0))) * sqrt(1.0 - ((v * v) * 3.0)))) / (1.0 - (v * v));
}

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. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied associate-/r*_binary640.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2021202 
(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)))))