Average Error: 0.5 → 0.6
Time: 6.4s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1}{t}}{\pi \cdot \sqrt{2}} - \left(2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + 6.625 \cdot \frac{{v}^{4}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)\]
\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{\frac{1}{t}}{\pi \cdot \sqrt{2}} - \left(2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + 6.625 \cdot \frac{{v}^{4}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\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) (* PI (sqrt 2.0)))
  (+
   (* 2.5 (/ (* v v) (* t (* PI (sqrt 2.0)))))
   (* 6.625 (/ (pow v 4.0) (* t (* PI (sqrt 2.0))))))))
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) / (((double) M_PI) * sqrt(2.0))) - ((2.5 * ((v * v) / (t * (((double) M_PI) * sqrt(2.0))))) + (6.625 * (pow(v, 4.0) / (t * (((double) M_PI) * sqrt(2.0))))));
}

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 0.5

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

  3. Applied add-cube-cbrt_binary64_21590.5

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

  4. Applied associate-*r*_binary64_20640.5

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

  5. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)} - \left(2.5 \cdot \frac{{v}^{2}}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + 6.625 \cdot \frac{{v}^{4}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)}\]

  6. Simplified0.6

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)} - \left(2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + 6.625 \cdot \frac{{v}^{4}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)}\]
  7. Using strategy rm

  8. Applied associate-/r*_binary64_20680.6

    \[\leadsto \color{blue}{\frac{\frac{1}{t}}{\pi \cdot \sqrt{2}}} - \left(2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + 6.625 \cdot \frac{{v}^{4}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)\]

  9. Final simplification0.6

    \[\leadsto \frac{\frac{1}{t}}{\pi \cdot \sqrt{2}} - \left(2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + 6.625 \cdot \frac{{v}^{4}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)\]

Reproduce

herbie shell --seed 2020356 
(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)))))