Average Error: 0.5 → 0.3
Time: 7.6s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\begin{array}{l} t_1 := \pi \cdot \sqrt{2}\\ \frac{\frac{1}{t_1} - 4 \cdot \mathsf{fma}\left(\frac{v}{\pi}, \frac{v}{\sqrt{2}}, \frac{{v}^{4}}{t_1}\right)}{t \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \end{array} \]
\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 := \pi \cdot \sqrt{2}\\
\frac{\frac{1}{t_1} - 4 \cdot \mathsf{fma}\left(\frac{v}{\pi}, \frac{v}{\sqrt{2}}, \frac{{v}^{4}}{t_1}\right)}{t \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\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 (* PI (sqrt 2.0))))
   (/
    (- (/ 1.0 t_1) (* 4.0 (fma (/ v PI) (/ v (sqrt 2.0)) (/ (pow v 4.0) t_1))))
    (* t (sqrt (fma (* v v) -3.0 1.0))))))
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 = ((double) M_PI) * sqrt(2.0);
	return ((1.0 / t_1) - (4.0 * fma((v / ((double) M_PI)), (v / sqrt(2.0)), (pow(v, 4.0) / t_1)))) / (t * sqrt(fma((v * v), -3.0, 1.0)));
}

Error

Bits error versus v

Bits error versus t

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 t around 0 0.4

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

  3. Applied associate-/r*_binary640.1

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

  4. Simplified0.1

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

  5. Applied sqrt-div_binary640.1

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

  6. Applied frac-times_binary640.1

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

  7. Simplified0.1

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

  8. Simplified0.1

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

  9. Taylor expanded in v around 0 0.3

    \[\leadsto \frac{\color{blue}{\frac{1}{\pi \cdot \sqrt{2}} - \left(4 \cdot \frac{{v}^{2}}{\pi \cdot \sqrt{2}} + 4 \cdot \frac{{v}^{4}}{\pi \cdot \sqrt{2}}\right)}}{t \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \]

  10. Simplified0.3

    \[\leadsto \frac{\color{blue}{\frac{1}{\pi \cdot \sqrt{2}} - 4 \cdot \mathsf{fma}\left(\frac{v}{\pi}, \frac{v}{\sqrt{2}}, \frac{{v}^{4}}{\pi \cdot \sqrt{2}}\right)}}{t \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \]

  11. Final simplification0.3

    \[\leadsto \frac{\frac{1}{\pi \cdot \sqrt{2}} - 4 \cdot \mathsf{fma}\left(\frac{v}{\pi}, \frac{v}{\sqrt{2}}, \frac{{v}^{4}}{\pi \cdot \sqrt{2}}\right)}{t \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -3, 1\right)}} \]

Reproduce

herbie shell --seed 2022125 
(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)))))