Average Error: 0.4 → 0.3
Time: 5.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{1}{t} \cdot \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \left(\pi - \left(v \cdot v\right) \cdot \pi\right)} \]
\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{1}{t} \cdot \frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)} \cdot \left(\pi - \left(v \cdot v\right) \cdot \pi\right)}

(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 t)
  (/
   (fma (* v v) -5.0 1.0)
   (* (sqrt (fma (* v v) -6.0 2.0)) (- PI (* (* v v) PI))))))
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 / t) * (fma((v * v), -5.0, 1.0) / (sqrt(fma((v * v), -6.0, 2.0)) * (((double) M_PI) - ((v * v) * ((double) M_PI)))));
}

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. Simplified0.5

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

  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. Applied add-cube-cbrt_binary640.4

    \[\leadsto \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t \cdot \left(\pi \cdot \sqrt{2 - 6 \cdot \color{blue}{\left(\left(\sqrt[3]{{v}^{2}} \cdot \sqrt[3]{{v}^{2}}\right) \cdot \sqrt[3]{{v}^{2}}\right)}} - \left({v}^{2} \cdot \pi\right) \cdot \sqrt{2 - 6 \cdot {v}^{2}}\right)} \]

  5. Applied associate-*r*_binary640.4

    \[\leadsto \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{t \cdot \left(\pi \cdot \sqrt{2 - \color{blue}{\left(6 \cdot \left(\sqrt[3]{{v}^{2}} \cdot \sqrt[3]{{v}^{2}}\right)\right) \cdot \sqrt[3]{{v}^{2}}}} - \left({v}^{2} \cdot \pi\right) \cdot \sqrt{2 - 6 \cdot {v}^{2}}\right)} \]

  6. Applied *-un-lft-identity_binary640.4

    \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(v, v \cdot -5, 1\right)}}{t \cdot \left(\pi \cdot \sqrt{2 - \left(6 \cdot \left(\sqrt[3]{{v}^{2}} \cdot \sqrt[3]{{v}^{2}}\right)\right) \cdot \sqrt[3]{{v}^{2}}} - \left({v}^{2} \cdot \pi\right) \cdot \sqrt{2 - 6 \cdot {v}^{2}}\right)} \]

  7. Applied times-frac_binary640.3

    \[\leadsto \color{blue}{\frac{1}{t} \cdot \frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi \cdot \sqrt{2 - \left(6 \cdot \left(\sqrt[3]{{v}^{2}} \cdot \sqrt[3]{{v}^{2}}\right)\right) \cdot \sqrt[3]{{v}^{2}}} - \left({v}^{2} \cdot \pi\right) \cdot \sqrt{2 - 6 \cdot {v}^{2}}}} \]

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2022097 
(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)))))