Average Error: 0.5 → 0.1
Time: 5.1s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right) \cdot \frac{\frac{1}{\pi \cdot \sqrt{2}}}{1 - v \cdot v}}{t} \cdot \sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}} \]
\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{\frac{1}{\pi \cdot \sqrt{2}}}{1 - v \cdot v}}{t} \cdot \sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}}

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

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. 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 add-sqr-sqrt_binary640.4

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

  5. Applied times-frac_binary640.3

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

  6. Simplified0.3

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

  7. Simplified0.3

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

  8. Applied associate-*l/_binary640.1

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

  9. Simplified0.1

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

  10. Applied div-inv_binary640.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

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)))))