Average Error: 0.4 → 0.1
Time: 3.6s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1 - v \cdot \left(v \cdot 5\right)}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)} \cdot \left(\left(1 - v \cdot v\right) \cdot \pi\right)}}{t}\]
\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{1 - v \cdot \left(v \cdot 5\right)}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)} \cdot \left(\left(1 - v \cdot v\right) \cdot \pi\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 (* v (* v 5.0)))
   (* (sqrt (* 2.0 (- 1.0 (* v (* v 3.0))))) (* (- 1.0 (* v v)) PI)))
  t))
double code(double v, double t) {
	return (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (((double) (((double) M_PI) * t)) * ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (3.0 * ((double) (v * v)))))))))))) * ((double) (1.0 - ((double) (v * v)))))));
}
↓
double code(double v, double t) {
	return ((((double) (1.0 - ((double) (v * ((double) (v * 5.0)))))) / ((double) (((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (v * ((double) (v * 3.0)))))))))) * ((double) (((double) (1.0 - ((double) (v * v)))) * ((double) M_PI)))))) / 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 Error: 0.4 bits

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

  2. SimplifiedError: 0.4 bits

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

  4. Applied associate-/r*Error: 0.3 bits

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

  5. SimplifiedError: 0.3 bits

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

  7. Applied *-un-lft-identityError: 0.3 bits

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

  8. Applied add-sqr-sqrtError: 0.3 bits

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

  9. Applied times-fracError: 0.3 bits

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

  10. Applied times-fracError: 0.3 bits

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

  11. SimplifiedError: 0.3 bits

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

  12. SimplifiedError: 0.3 bits

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

  14. Applied associate-*l/Error: 0.1 bits

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

  15. SimplifiedError: 0.1 bits

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

  16. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020203 
(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)))))