Average Error: 0.4 → 0.1
Time: 2.7s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{\pi \cdot \left(1 - v \cdot v\right)}}{t}\]
\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{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{\pi \cdot \left(1 - v \cdot v\right)}}{t}
(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 (* 5.0 (* v v))) (sqrt (+ 2.0 (* v (* v -6.0)))))
   (* PI (- 1.0 (* v v))))
  t))
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))) / sqrt(2.0 + (v * (v * -6.0)))) / (((double) M_PI) * (1.0 - (v * v)))) / t;
}

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

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

  4. Applied associate-/r*_binary640.4

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

  5. Simplified0.4

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

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

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

  8. Applied sqrt-prod_binary640.4

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

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

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

  10. Applied times-frac_binary640.4

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

  11. Applied times-frac_binary640.3

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

  12. Simplified0.3

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

  14. Applied associate-*l/_binary640.1

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2020224 
(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)))))