Average Error: 0.5 → 0.1
Time: 5.0s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)}}{t}\]
\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{1 - 5 \cdot \left(v \cdot v\right)}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)}}{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 (* 5.0 (* v v)))
   (* (- 1.0 (* v v)) (* PI (sqrt (- 2.0 (* (* v v) 6.0))))))
  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 - (5.0 * (v * v))) / ((1.0 - (v * v)) * (((double) M_PI) * sqrt(2.0 - ((v * v) * 6.0))))) / 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 0.5

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

  2. Simplified0.5

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  6. Applied *-un-lft-identity_binary640.4

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

  7. Applied times-frac_binary640.3

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

  8. Simplified0.3

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

  10. Applied associate-*l/_binary640.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))))