Average Error: 0.5 → 0.3
Time: 4.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{\frac{1}{\pi}}{\sqrt{2}} \cdot \frac{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{t}}{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{1 - v \cdot v}\]
\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{\frac{1}{\pi}}{\sqrt{2}} \cdot \frac{\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{t}}{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{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
 (*
  (*
   (/ (/ 1.0 PI) (sqrt 2.0))
   (/ (/ (sqrt (- 1.0 (* (* v v) 5.0))) t) (sqrt (- 1.0 (* (* v v) 3.0)))))
  (/ (sqrt (- 1.0 (* (* v v) 5.0))) (- 1.0 (* v 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 / ((double) M_PI)) / sqrt(2.0)) * ((sqrt(1.0 - ((v * v) * 5.0)) / t) / sqrt(1.0 - ((v * v) * 3.0)))) * (sqrt(1.0 - ((v * v) * 5.0)) / (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 0.5

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

  3. Applied add-sqr-sqrt_binary640.5

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

  4. Applied times-frac_binary640.5

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

  5. Simplified0.5

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

  6. Simplified0.5

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

  8. Applied associate-/r*_binary640.5

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

  10. Applied sqrt-prod_binary640.5

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

  11. Applied *-un-lft-identity_binary640.5

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

  12. Applied sqrt-prod_binary640.5

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

  13. Applied times-frac_binary640.4

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

  14. Applied times-frac_binary640.3

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

  15. Simplified0.3

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

  16. Simplified0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2020260 
(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)))))