Average Error: 0.4 → 0.4
Time: 7.3s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\begin{array}{l} t_1 := \sqrt{2 - 6 \cdot {v}^{2}}\\ \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t \cdot \left(\pi \cdot t_1 - t_1 \cdot \left(\pi \cdot {v}^{2}\right)\right)} \end{array} \]
\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 := \sqrt{2 - 6 \cdot {v}^{2}}\\
\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t \cdot \left(\pi \cdot t_1 - t_1 \cdot \left(\pi \cdot {v}^{2}\right)\right)}
\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 (sqrt (- 2.0 (* 6.0 (pow v 2.0))))))
   (/ (fma v (* v -5.0) 1.0) (* t (- (* PI t_1) (* t_1 (* PI (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) {
	double t_1 = sqrt(2.0 - (6.0 * pow(v, 2.0)));
	return fma(v, (v * -5.0), 1.0) / (t * ((((double) M_PI) * t_1) - (t_1 * (((double) M_PI) * pow(v, 2.0)))));
}

Error

Bits error versus v

Bits error versus t

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

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

  3. Taylor expanded in t around 0 0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2022093 
(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)))))