Average Error: 0.4 → 0.4
Time: 6.1s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\left(1 - v \cdot \left(v \cdot 5\right)\right) \cdot \frac{\frac{\sqrt[3]{1}}{\pi} \cdot \frac{\sqrt[3]{1}}{t}}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}}\right) \cdot \frac{\sqrt[3]{1}}{1 - v \cdot v}\]
\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)}
↓
\left(\left(1 - v \cdot \left(v \cdot 5\right)\right) \cdot \frac{\frac{\sqrt[3]{1}}{\pi} \cdot \frac{\sqrt[3]{1}}{t}}{\sqrt{2 \cdot \left(1 - v \cdot \left(v \cdot 3\right)\right)}}\right) \cdot \frac{\sqrt[3]{1}}{1 - v \cdot v}
(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)))
   (/
    (* (/ (cbrt 1.0) PI) (/ (cbrt 1.0) t))
    (sqrt (* 2.0 (- 1.0 (* v (* v 3.0)))))))
  (/ (cbrt 1.0) (- 1.0 (* v v)))))
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) (((double) (((double) (1.0 - ((double) (v * ((double) (v * 5.0)))))) * (((double) ((((double) cbrt(1.0)) / ((double) M_PI)) * (((double) cbrt(1.0)) / t))) / ((double) sqrt(((double) (2.0 * ((double) (1.0 - ((double) (v * ((double) (v * 3.0))))))))))))) * (((double) cbrt(1.0)) / ((double) (1.0 - ((double) (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 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. Using strategy rm

  3. Applied clear-numError: 0.4 bits

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

  4. SimplifiedError: 0.4 bits

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

  6. Applied add-cube-cbrtError: 0.4 bits

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

  7. Applied times-fracError: 0.4 bits

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

  8. SimplifiedError: 0.4 bits

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

  10. Applied add-sqr-sqrtError: 0.4 bits

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

  11. Applied times-fracError: 0.4 bits

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

  12. Applied associate-*r*Error: 0.4 bits

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

  14. Applied associate-*r/Error: 0.4 bits

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

  15. SimplifiedError: 0.4 bits

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

  16. Final simplificationError: 0.4 bits

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

Reproduce

herbie shell --seed 2020204 
(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)))))