Average Error: 0.4 → 0.4
Time: 7.6s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{\pi \cdot \sqrt{2}}}{t} - 2.5 \cdot \frac{v \cdot v}{\left(\pi \cdot \sqrt{2}\right) \cdot t}\]
\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}{\pi \cdot \sqrt{2}}}{t} - 2.5 \cdot \frac{v \cdot v}{\left(\pi \cdot \sqrt{2}\right) \cdot t}
(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 (* PI (sqrt 2.0))) t)
  (* 2.5 (/ (* v v) (* (* PI (sqrt 2.0)) t)))))
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 / (((double) M_PI) * sqrt(2.0))) / t) - (2.5 * ((v * v) / ((((double) M_PI) * sqrt(2.0)) * t)));
}

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 0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)} - 2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)} - 2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)}}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt_binary64_18050.6

    \[\leadsto \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} - 2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\]

  6. Applied times-frac_binary64_17890.6

    \[\leadsto \color{blue}{\frac{\sqrt{1}}{t} \cdot \frac{\sqrt{1}}{\pi \cdot \sqrt{2}}} - 2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\]

  7. Simplified0.6

    \[\leadsto \color{blue}{\frac{1}{t}} \cdot \frac{\sqrt{1}}{\pi \cdot \sqrt{2}} - 2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\]

  8. Simplified0.6

    \[\leadsto \frac{1}{t} \cdot \color{blue}{\frac{1}{\pi \cdot \sqrt{2}}} - 2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\]
  9. Using strategy rm

  10. Applied associate-*l/_binary64_17260.4

    \[\leadsto \color{blue}{\frac{1 \cdot \frac{1}{\pi \cdot \sqrt{2}}}{t}} - 2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\]

  11. Final simplification0.4

    \[\leadsto \frac{\frac{1}{\pi \cdot \sqrt{2}}}{t} - 2.5 \cdot \frac{v \cdot v}{\left(\pi \cdot \sqrt{2}\right) \cdot t}\]

Reproduce

herbie shell --seed 2021028 
(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)))))