Average Error: 0.4 → 0.1
Time: 5.9s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \sqrt{2}\right)}}{t} \cdot \sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}}\]
\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 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \sqrt{2}\right)}}{t} \cdot \sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}}
(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 (* (* v v) 5.0)) (* (- 1.0 (* v v)) (* PI (sqrt 2.0)))) t)
  (sqrt (/ 1.0 (- 1.0 (* (* v v) 3.0))))))
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 - ((v * v) * 5.0)) / ((1.0 - (v * v)) * (((double) M_PI) * sqrt(2.0)))) / t) * sqrt(1.0 / (1.0 - ((v * v) * 3.0)));
}

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. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied add-sqr-sqrt_binary64_18050.4

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

  6. Applied times-frac_binary64_17890.3

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

  8. Applied associate-*l/_binary64_17260.2

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020357 
(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)))))