Average Error: 0.4 → 0.1
Time: 7.1s
Precision: binary64
Cost: 14656
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}} \cdot \frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \sqrt{2}\right)}}{t}\]
\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)}
↓
\sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}} \cdot \frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \sqrt{2}\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
 (*
  (sqrt (/ 1.0 (- 1.0 (* (* v v) 3.0))))
  (/ (/ (- 1.0 (* (* v v) 5.0)) (* (- 1.0 (* v v)) (* PI (sqrt 2.0)))) 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 sqrt(1.0 / (1.0 - ((v * v) * 3.0))) * (((1.0 - ((v * v) * 5.0)) / ((1.0 - (v * v)) * (((double) M_PI) * sqrt(2.0)))) / t);
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.4
Cost14656
\[\sqrt{\frac{1}{1 - \left(v \cdot v\right) \cdot 3}} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{\left(\pi \cdot \sqrt{2}\right) \cdot \left(\left(1 - v \cdot v\right) \cdot t\right)}\]
Alternative 2
Error0.4
Cost8256
\[\frac{1}{\left(\pi \cdot t\right) \cdot \sqrt{\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot 2}} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v}\]
Alternative 3
Error0.4
Cost8128
\[\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(\pi \cdot t\right) \cdot \sqrt{\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot 2}}}{1 - v \cdot v}\]
Alternative 4
Error0.4
Cost8128
\[\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{\left(1 - \left(v \cdot v\right) \cdot 3\right) \cdot 2}\right)}\]
Alternative 5
Error0.4
Cost8000
\[\frac{1 - \left(v \cdot v\right) \cdot 5}{\sqrt{2 + v \cdot \left(v \cdot -6\right)} \cdot \left(\left(1 - v \cdot v\right) \cdot \left(\pi \cdot t\right)\right)}\]
Alternative 6
Error1.0
Cost7616
\[\frac{1 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\sqrt{2} \cdot \left(\pi \cdot t\right)\right)}\]
Alternative 7
Error1.0
Cost6848
\[\frac{1}{\sqrt{2} \cdot \left(\pi \cdot t\right)}\]
Alternative 8
Error60.6
Cost385
\[\begin{array}{l} \mathbf{if}\;t \leq 9.0532805922654 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 9
Error61.6
Cost64
\[0\]
Alternative 10
Error61.7
Cost64
\[1\]

Error

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. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied *-un-lft-identity_binary64_17830.4

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

  6. Applied times-frac_binary64_17890.3

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

  8. Applied associate-*l/_binary64_17260.1

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

  9. Simplified0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2021044 
(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)))))