Average Error: 0.4 → 0.3
Time: 5.6s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}}{1 - v \cdot v}\]
\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{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}}}{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
 (/
  (/
   (/ (- 1.0 (* (* v v) 5.0)) PI)
   (* t (sqrt (* 2.0 (- 1.0 (* (* v v) 3.0))))))
  (- 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 (((1.0 - ((v * v) * 5.0)) / ((double) M_PI)) / (t * sqrt(2.0 * (1.0 - ((v * v) * 3.0))))) / (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 associate-*l*_binary640.4

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

  5. Applied associate-/r*_binary640.4

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

  6. Simplified0.4

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

  8. Applied add-sqr-sqrt_binary640.5

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

  9. Applied times-frac_binary640.4

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

  11. Applied associate-*l/_binary640.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2020275 
(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)))))