Average Error: 0.4 → 0.3
Time: 4.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{t} \cdot \frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi \cdot \left(1 - v \cdot v\right)}}{\sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}\]
\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 (((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) ((1.0 / t) * ((((double) (1.0 - ((double) (((double) (v * v)) * 5.0)))) / ((double) (((double) M_PI) * ((double) (1.0 - ((double) (v * v))))))) / ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (((double) (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 Error: 0.4 bits

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

  3. Applied associate-*l*Error: 0.4 bits

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

  5. Applied clear-numError: 0.4 bits

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

  6. SimplifiedError: 0.4 bits

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

  8. Applied *-un-lft-identityError: 0.4 bits

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

  9. Applied *-un-lft-identityError: 0.4 bits

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

  10. Applied times-fracError: 0.4 bits

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

  11. Applied times-fracError: 0.4 bits

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

  12. Applied add-sqr-sqrtError: 0.4 bits

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

  13. Applied times-fracError: 0.4 bits

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

  14. SimplifiedError: 0.4 bits

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

  15. SimplifiedError: 0.4 bits

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

  17. Applied *-un-lft-identityError: 0.4 bits

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

  18. Applied times-fracError: 0.4 bits

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

  19. Applied associate-*l*Error: 0.3 bits

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

  20. SimplifiedError: 0.3 bits

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

  21. Final simplificationError: 0.3 bits

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

Reproduce

herbie shell --seed 2020205 
(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)))))