Average Error: 0.4 → 0.4
Time: 11.1s
Precision: binary64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1 - v \cdot v} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{t \cdot \pi}\right)\]
\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)}
↓
\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1 - v \cdot v} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{t \cdot \pi}\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
 (*
  (- 1.0 (* 5.0 (* v v)))
  (*
   (/ (* (cbrt 1.0) (cbrt 1.0)) (- 1.0 (* v v)))
   (/ (/ (cbrt 1.0) (sqrt (- 2.0 (* (* v v) 6.0)))) (* t PI)))))
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))) * (((cbrt(1.0) * cbrt(1.0)) / (1.0 - (v * v))) * ((cbrt(1.0) / sqrt(2.0 - ((v * v) * 6.0))) / (t * ((double) M_PI))));
}

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

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

  4. Applied div-inv_binary64_21210.4

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

  5. Simplified0.4

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

  7. Applied *-un-lft-identity_binary64_21240.4

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

  8. Applied sqrt-prod_binary64_21400.4

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

  9. Applied add-cube-cbrt_binary64_21590.4

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

  10. Applied times-frac_binary64_21300.4

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

  11. Applied times-frac_binary64_21300.4

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

  12. Final simplification0.4

    \[\leadsto \left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1 - v \cdot v} \cdot \frac{\frac{\sqrt[3]{1}}{\sqrt{2 - \left(v \cdot v\right) \cdot 6}}}{t \cdot \pi}\right)\]

Reproduce

herbie shell --seed 2021015 
(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)))))