Average Error: 0.4 → 0.1
Time: 3.4s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\sqrt{1 - v \cdot \left(v \cdot 5\right)}}{\pi \cdot \sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}}}{t} \cdot \frac{\sqrt{1 - v \cdot \left(v \cdot 5\right)}}{1 - v \cdot v}\]
\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)}
↓
\frac{\frac{\sqrt{1 - v \cdot \left(v \cdot 5\right)}}{\pi \cdot \sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}}}{t} \cdot \frac{\sqrt{1 - v \cdot \left(v \cdot 5\right)}}{1 - v \cdot v}
(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
 (*
  (/
   (/
    (sqrt (- 1.0 (* v (* v 5.0))))
    (* PI (sqrt (* 2.0 (- 1.0 (* v (* v 3.0)))))))
   t)
  (/ (sqrt (- 1.0 (* v (* v 5.0)))) (- 1.0 (* v 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) sqrt(((double) (1.0 - ((double) (v * ((double) (v * 5.0)))))))) / ((double) (((double) M_PI) * ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (v * ((double) (v * 3.0))))))))))))) / t) * (((double) sqrt(((double) (1.0 - ((double) (v * ((double) (v * 5.0)))))))) / ((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 0.4

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

  3. Applied add-sqr-sqrt0.4

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

  4. Applied times-frac0.4

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

  5. Simplified0.4

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.3

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

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

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

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

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

  12. Applied sqrt-prod0.3

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

  13. Applied times-frac0.3

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

  14. Applied times-frac0.3

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

  15. Simplified0.3

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

  16. Simplified0.3

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

  18. Applied associate-*l/0.1

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

  19. Simplified0.1

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

  20. Final simplification0.1

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

Reproduce

herbie shell --seed 2020198 
(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)))))