Average Error: 0.5 → 0.5
Time: 4.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\begin{array}{l} t_1 := \pi \cdot \sqrt{2}\\ \frac{\frac{1}{t_1}}{t} + \frac{v \cdot v}{t_1 \cdot t} \cdot -2.5 \end{array} \]
\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)}
↓
\begin{array}{l}
t_1 := \pi \cdot \sqrt{2}\\
\frac{\frac{1}{t_1}}{t} + \frac{v \cdot v}{t_1 \cdot t} \cdot -2.5
\end{array}

(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
 (let* ((t_1 (* PI (sqrt 2.0))))
   (+ (/ (/ 1.0 t_1) t) (* (/ (* v v) (* t_1 t)) -2.5))))
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) {
	double t_1 = ((double) M_PI) * sqrt(2.0);
	return ((1.0 / t_1) / t) + (((v * v) / (t_1 * t)) * -2.5);
}

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 0.5

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

  2. Taylor expanded in v around 0 0.8

    \[\leadsto \color{blue}{\left(\frac{1}{\pi \cdot \left(t \cdot \sqrt{2}\right)} + 1.5 \cdot \frac{{v}^{2}}{\pi \cdot \left(t \cdot \sqrt{2}\right)}\right) - 4 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}} \]

  3. Simplified0.8

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

  4. Applied add-cube-cbrt_binary640.8

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} \cdot -2.5 \]

  5. Applied times-frac_binary640.7

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{t} \cdot \frac{\sqrt[3]{1}}{\pi \cdot \sqrt{2}}} + \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} \cdot -2.5 \]

  6. Simplified0.7

    \[\leadsto \color{blue}{\frac{1}{t}} \cdot \frac{\sqrt[3]{1}}{\pi \cdot \sqrt{2}} + \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} \cdot -2.5 \]

  7. Simplified0.7

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

  8. Applied associate-*l/_binary640.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2021215 
(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)))))