Average Error: 0.4 → 0.3
Time: 7.3s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{t} \cdot \frac{\frac{-1 + \left(v \cdot v\right) \cdot 5}{\pi \cdot \left(-1 + v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}\]
\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{1}{t} \cdot \frac{\frac{-1 + \left(v \cdot v\right) \cdot 5}{\pi \cdot \left(-1 + v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\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 t)
  (/
   (/ (+ -1.0 (* (* v v) 5.0)) (* PI (+ -1.0 (* v v))))
   (sqrt (* 2.0 (- 1.0 (* (* v v) 3.0)))))))
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 / t) * (((-1.0 + ((v * v) * 5.0)) / (((double) M_PI) * (-1.0 + (v * v)))) / sqrt(2.0 * (1.0 - ((v * v) * 3.0))));
}

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. Simplified0.4

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

  7. Applied associate-/r*_binary640.3

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

  8. Simplified0.3

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

  10. Applied *-un-lft-identity_binary640.3

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

  11. Applied *-un-lft-identity_binary640.3

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

  12. Applied times-frac_binary640.3

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

  13. Applied times-frac_binary640.4

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

  14. Simplified0.4

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

  15. Simplified0.4

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

  17. Applied *-un-lft-identity_binary640.4

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

  18. Applied times-frac_binary640.4

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

  19. Applied associate-*l*_binary640.3

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

  20. Simplified0.3

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

  21. Final simplification0.3

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

Reproduce

herbie shell --seed 2021175 
(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)))))