Average Error: 0.5 → 0.1
Time: 6.2s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}{\sqrt{2}}}{t \cdot \left(1 - v \cdot v\right)}\]
\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{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}{\sqrt{2}}}{t \cdot \left(1 - v \cdot v\right)}
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))) / ((double) M_PI)) / sqrt((1.0 - (3.0 * (v * v))))) / sqrt(2.0)) / (t * (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 0.5

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

  3. Applied associate-*l*0.5

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

  4. Applied associate-*l*0.5

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

  5. Applied associate-/r*0.3

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

  7. Applied *-commutative0.3

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

  8. Applied associate-*l*0.3

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

  9. Applied associate-/r*0.1

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

  11. Applied *-commutative0.1

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

  12. Applied sqrt-prod0.1

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

  13. Applied associate-/r*0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2020078 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))