Average Error: 0.5 → 0.1
Time: 5.5s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right) \cdot \frac{\sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}}}{\pi \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(v, -v, 1\right)\right)}}{t} \]
\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{\mathsf{fma}\left(v, v \cdot -5, 1\right) \cdot \frac{\sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}}}{\pi \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(v, -v, 1\right)\right)}}{t}

(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)
   (/
    (sqrt (/ 1.0 (- 1.0 (* (* v v) 3.0))))
    (* PI (* (sqrt 2.0) (fma v (- v) 1.0)))))
  t))
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) * (sqrt(1.0 / (1.0 - ((v * v) * 3.0))) / (((double) M_PI) * (sqrt(2.0) * fma(v, -v, 1.0))))) / t;
}

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. Simplified0.4

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

  4. Applied associate-/r*_binary640.3

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

  5. Simplified0.3

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

  6. Applied div-inv_binary640.3

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

  7. Applied associate-*l*_binary640.3

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

  8. Simplified0.3

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

  9. Applied associate-*l/_binary640.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2022082 
(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)))))