Average Error: 0.4 → 0.1
Time: 11.0s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\mathsf{fma}\left(v, -v, 1\right) \cdot \left(\pi \cdot \sqrt{2}\right)}}{t} \cdot \sqrt{\frac{1}{1 - 3 \cdot {v}^{2}}} \]
\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{\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}}}

(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
 (*
  (/ (/ (fma (* v v) -5.0 1.0) (* (fma v (- v) 1.0) (* PI (sqrt 2.0)))) t)
  (sqrt (/ 1.0 (- 1.0 (* 3.0 (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) {
	return ((fma((v * v), -5.0, 1.0) / (fma(v, -v, 1.0) * (((double) M_PI) * sqrt(2.0)))) / t) * sqrt(1.0 / (1.0 - (3.0 * 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. 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. Final simplification0.1

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

Reproduce

herbie shell --seed 2022068 
(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)))))