Average Error: 0.4 → 0.3
Time: 5.3s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
↓
\[\frac{\frac{\left(v \cdot v\right) \cdot 5 - 1}{\pi}}{\left(v \cdot v + -1\right) \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}\right)} \]
\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{\left(v \cdot v\right) \cdot 5 - 1}{\pi}}{\left(v \cdot v + -1\right) \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - \left(v \cdot v\right) \cdot 3\right)}\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
 (/
  (/ (- (* (* v v) 5.0) 1.0) PI)
  (* (+ (* v v) -1.0) (* t (sqrt (* 2.0 (- 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 ((((v * v) * 5.0) - 1.0) / ((double) M_PI)) / (((v * v) + -1.0) * (t * sqrt(2.0 * (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. Using strategy rm

  3. Applied frac-2neg_binary640.4

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

  4. Simplified0.4

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

  5. Simplified0.4

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

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

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

  8. Applied times-frac_binary640.4

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

  9. Simplified0.4

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

  11. Applied associate-*r/_binary640.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2021198 
(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)))))