Average Error: 0.4 → 0.3
Time: 11.1s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{t} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{\pi \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{2 - \left(v \cdot v\right) \cdot 6}\right)}\]
\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}{t} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{\pi \cdot \left(\left(1 - v \cdot v\right) \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))
   (* PI (* (- 1.0 (* v v)) (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)) / (((double) M_PI) * ((1.0 - (v * v)) * 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. Using strategy rm

  4. Applied associate-/r*_binary64_17270.4

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

  5. Simplified0.4

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

  7. Applied associate-/l/_binary64_17300.4

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

  8. Simplified0.4

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

  10. Applied *-un-lft-identity_binary64_17830.4

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

  11. Applied times-frac_binary64_17890.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2021098 
(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)))))