Average Error: 0.4 → 0.4
Time: 5.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(t \cdot \left(\pi \cdot \sqrt{2 - \sqrt{6} \cdot \left(v \cdot \left(v \cdot \sqrt{6}\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\right)}\]
\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 - 5 \cdot \left(v \cdot v\right)}{\left(t \cdot \left(\pi \cdot \sqrt{2 - \sqrt{6} \cdot \left(v \cdot \left(v \cdot \sqrt{6}\right)\right)}\right)\right) \cdot \left(1 - v \cdot v\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 (* 5.0 (* v v)))
  (*
   (* t (* PI (sqrt (- 2.0 (* (sqrt 6.0) (* v (* v (sqrt 6.0))))))))
   (- 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 - (5.0 * (v * v))) / ((t * (((double) M_PI) * sqrt(2.0 - (sqrt(6.0) * (v * (v * sqrt(6.0))))))) * (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. 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*_binary640.4

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

  5. Simplified0.4

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

  7. Applied add-sqr-sqrt_binary640.4

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

  8. Applied associate-*r*_binary640.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2021126 
(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)))))