Average Error: 0.4 → 0.3
Time: 7.6s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{-1 + \left(v \cdot v\right) \cdot 5}{\pi}}{\left(t \cdot \sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}\right) \cdot \left(-1 + {v}^{4}\right)} \cdot \left(v \cdot v + 1\right)\]
\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{-1 + \left(v \cdot v\right) \cdot 5}{\pi}}{\left(t \cdot \sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}\right) \cdot \left(-1 + {v}^{4}\right)} \cdot \left(v \cdot v + 1\right)
(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 (* (* v v) 5.0)) PI)
   (* (* t (sqrt (* 2.0 (- 1.0 (* (* v v) 3.0))))) (+ -1.0 (pow v 4.0))))
  (+ (* v v) 1.0)))
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 + ((v * v) * 5.0)) / ((double) M_PI)) / ((t * sqrt(2.0 * (1.0 - ((v * v) * 3.0)))) * (-1.0 + pow(v, 4.0)))) * ((v * v) + 1.0);
}

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 frac-2neg_binary640.4

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

  4. Simplified0.4

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

  5. Simplified0.5

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

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

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

  8. Applied times-frac_binary640.4

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

  9. Simplified0.4

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

  11. Applied flip-+_binary640.4

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

  12. Applied associate-*l/_binary640.4

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

  13. Applied associate-/r/_binary640.4

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

  14. Applied associate-*r*_binary640.4

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2021174 
(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)))))