Average Error: 0.5 → 0.3
Time: 4.8s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{1}{t} \cdot \frac{1}{\pi \cdot \sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}}\right) \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]
\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)}
↓
\left(\frac{1}{t} \cdot \frac{1}{\pi \cdot \sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}}\right) \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}
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 / t) * (1.0 / ((double) (((double) M_PI) * ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (v * ((double) (v * 3.0))))))))))))))) * (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (1.0 - ((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 *-un-lft-identity0.5

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

  4. Applied times-frac0.5

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

  5. Simplified0.5

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

  7. Applied associate-/r*0.3

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

  9. Applied *-un-lft-identity0.3

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

  10. Applied add-sqr-sqrt0.3

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

  11. Applied times-frac0.3

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

  12. Applied times-frac0.3

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

  13. Simplified0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

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)))))