Average Error: 0.4 → 0.3
Time: 7.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \pi} \cdot \frac{\frac{1}{t}}{1 - v \cdot v}\]
\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 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \pi} \cdot \frac{\frac{1}{t}}{1 - v \cdot v}
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 - (5.0 * (v * v))) / (sqrt((2.0 * (1.0 - (3.0 * (v * v))))) * ((double) M_PI))) * ((1.0 / t) / (1.0 - (v * v))));
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied flip--0.4

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

  4. Applied associate-/l/0.4

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

  5. Simplified0.4

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

  7. Applied associate-/r*0.4

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

  8. Simplified0.4

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

  10. Applied associate-*r/0.3

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

  11. Simplified0.3

    \[\leadsto \frac{\frac{\color{blue}{\frac{\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot 1}{\pi}}}{t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}\]
  12. Using strategy rm

  13. Applied div-inv0.4

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

  14. Applied times-frac0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020103 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))