Average Error: 0.4 → 0.3
Time: 23.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{t} \cdot \frac{\frac{\sqrt[3]{1}}{\pi}}{\frac{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}}}\]
\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)}
↓
\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{t} \cdot \frac{\frac{\sqrt[3]{1}}{\pi}}{\frac{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}{\frac{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
 (*
  (/ (* (cbrt 1.0) (cbrt 1.0)) t)
  (/
   (/ (cbrt 1.0) PI)
   (/
    (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))
    (/ (- 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 ((cbrt(1.0) * cbrt(1.0)) / t) * ((cbrt(1.0) / ((double) M_PI)) / (sqrt(2.0 * (1.0 - (3.0 * (v * v)))) / ((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 0.4

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

  3. Applied clear-num_binary64_14410.4

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

  4. Simplified0.4

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

  6. Applied *-un-lft-identity_binary64_14420.4

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

  7. Applied *-un-lft-identity_binary64_14420.4

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

  8. Applied times-frac_binary64_14480.4

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

  9. Applied times-frac_binary64_14480.4

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

  10. Applied associate-/r*_binary64_13860.4

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

  11. Simplified0.4

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

  13. Applied *-un-lft-identity_binary64_14420.4

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

  14. Applied *-un-lft-identity_binary64_14420.4

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

  15. Applied times-frac_binary64_14480.4

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

  16. Applied add-cube-cbrt_binary64_14770.4

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

  17. Applied times-frac_binary64_14480.4

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

  18. Applied times-frac_binary64_14480.3

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

  19. Final simplification0.3

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

Reproduce

herbie shell --seed 2021032 
(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)))))