Average Error: 0.4 → 0.3
Time: 11.7s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{\frac{t}{\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)}}}\]
\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}{\frac{t}{\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)}}}
(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
   (/
    (- 1.0 (* (* v v) 5.0))
    (* (- 1.0 (* v v)) (* PI (sqrt (- 2.0 (* (* v v) 6.0)))))))))
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 / ((1.0 - ((v * v) * 5.0)) / ((1.0 - (v * v)) * (((double) M_PI) * sqrt(2.0 - ((v * v) * 6.0))))));
}

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 0.4

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

  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. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  6. Applied clear-num_binary64_17820.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2021024 
(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)))))