Average Error: 0.5 → 0.1
Time: 9.4s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}}{\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}}{t} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]
\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{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}}{\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}}{t} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}
double code(double v, double t) {
	return (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (((double) (((double) M_PI) * t)) * ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))))) * ((double) (1.0 - ((double) (v * v)))))));
}
↓
double code(double v, double t) {
	return ((double) ((((((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (1.0 - ((double) (v * v))))) / ((double) (((double) M_PI) * ((double) sqrt(((double) (((double) (((double) (1.0 * 1.0)) - ((double) (((double) pow(v, 4.0)) * ((double) (3.0 * 3.0)))))) * 2.0))))))) / t) * ((double) sqrt(((double) (1.0 + ((double) (3.0 * ((double) (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 0.5

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

  3. Applied flip--0.5

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

  4. Applied associate-*r/0.5

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

  5. Applied sqrt-div0.5

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

  6. Applied associate-*r/0.5

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

  7. Applied associate-*l/0.5

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

  8. Applied associate-/r/0.5

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

  9. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}}{t \cdot \left(\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}\right)}} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]
  10. Using strategy rm

  11. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{1 \cdot \left(1 - v \cdot v\right)}}}{t \cdot \left(\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}\right)} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

  12. Applied add-sqr-sqrt0.4

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}{1 \cdot \left(1 - v \cdot v\right)}}{t \cdot \left(\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}\right)} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

  13. Applied times-frac0.4

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}}}{t \cdot \left(\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}\right)} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

  14. Applied times-frac0.3

    \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1}}{t} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}}{\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}\right)} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

  15. Simplified0.3

    \[\leadsto \left(\color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t}} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}}{\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]
  16. Using strategy rm

  17. Applied associate-*l/0.1

    \[\leadsto \color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}}{\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}}{t}} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

  18. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}}{\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}}}{t} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

  19. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}}{\pi \cdot \sqrt{\left(1 \cdot 1 - {v}^{4} \cdot \left(3 \cdot 3\right)\right) \cdot 2}}}{t} \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\]

Reproduce

herbie shell --seed 2020182 
(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)))))