Average Error: 0.4 → 0.1
Time: 12.3s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{1 - v \cdot v}}{\pi}}{t}\]
\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{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{1 - v \cdot v}}{\pi}}{t}
(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)) (sqrt (+ 2.0 (* v (* v -6.0)))))
    (- 1.0 (* v v)))
   PI)
  t))
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)) / sqrt(2.0 + (v * (v * -6.0)))) / (1.0 - (v * v))) / ((double) M_PI)) / t;
}

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. Simplified0.4

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

  4. Applied associate-/r*_binary64_17080.4

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

  5. Simplified0.4

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

  7. Applied div-inv_binary64_17590.4

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

  8. Applied times-frac_binary64_17680.5

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

  10. Applied *-un-lft-identity_binary64_17620.5

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

  11. Applied times-frac_binary64_17680.4

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

  12. Applied associate-*l*_binary64_17050.4

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

  13. Simplified0.3

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

  15. Applied associate-*r/_binary64_17060.1

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

  16. Simplified0.1

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

  17. Final simplification0.1

    \[\leadsto \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{1 - v \cdot v}}{\pi}}{t}\]

Reproduce

herbie shell --seed 2020288 
(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)))))