Average Error: 0.5 → 0.1
Time: 4.9min
Precision: binary64
Cost: 40768
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{e^{\log \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216}}\right)}}{\pi}}{t} \cdot \frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v}\]
\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)}
↓
\frac{\frac{e^{\log \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216}}\right)}}{\pi}}{t} \cdot \frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{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
 (*
  (/
   (/
    (exp
     (log (/ (- 1.0 (* 5.0 (* v v))) (sqrt (+ 8.0 (* (pow v 6.0) -216.0))))))
    PI)
   t)
  (/
   (sqrt (+ 4.0 (- (* (pow v 4.0) 36.0) (* (* v v) -12.0))))
   (- 1.0 (* v v)))))
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 ((exp(log((1.0 - (5.0 * (v * v))) / sqrt(8.0 + (pow(v, 6.0) * -216.0)))) / ((double) M_PI)) / t) * (sqrt(4.0 + ((pow(v, 4.0) * 36.0) - ((v * v) * -12.0))) / (1.0 - (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

Alternatives

Alternative 1
Error0.1
Cost27968
\[\frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v} \cdot \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216} \cdot \pi}}{t}\]
Alternative 2
Error0.1
Cost27968
\[\frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v} \cdot \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216}}}{\pi}}{t}\]
Alternative 3
Error0.5
Cost27648
\[\frac{1}{t} \cdot \frac{1}{\pi \cdot \sqrt{2}} - \left(2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + 6.625 \cdot \frac{{v}^{4}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)\]
Alternative 4
Error0.5
Cost27520
\[\frac{\frac{1}{t}}{\pi \cdot \sqrt{2}} - \left(2.5 \cdot \frac{v \cdot v}{t \cdot \left(\pi \cdot \sqrt{2}\right)} + 6.625 \cdot \frac{{v}^{4}}{t \cdot \left(\pi \cdot \sqrt{2}\right)}\right)\]
Alternative 5
Error0.4
Cost14720
\[\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{\left(\pi \cdot t\right) \cdot \left(1 - {v}^{4}\right)} \cdot \left(1 + v \cdot v\right)\]
Alternative 6
Error0.4
Cost8128
\[\frac{\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \frac{1}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot t\right)}\]
Alternative 7
Error0.4
Cost8000
\[\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{2 + v \cdot \left(v \cdot -6\right)}}}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot t\right)}\]
Alternative 8
Error0.8
Cost6848
\[\frac{\frac{1}{\frac{\pi}{\sqrt{0.5}}}}{t}\]
Alternative 9
Error1.4
Cost6720
\[\frac{\frac{\sqrt{0.5}}{\pi}}{t}\]
Alternative 10
Error1.4
Cost6720
\[\frac{\sqrt{0.5}}{\pi \cdot t}\]
Alternative 11
Error60.5
Cost385
\[\begin{array}{l} \mathbf{if}\;t \leq 8.5714315881126 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 12
Error61.5
Cost64
\[0\]

Error

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

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

  4. Applied associate-/r*_binary64_17270.4

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

  5. Simplified0.4

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

  7. Applied flip3-+_binary64_17860.4

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

  8. Applied sqrt-div_binary64_18000.4

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

  9. Applied associate-/r/_binary64_17290.4

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

  10. Applied times-frac_binary64_17890.4

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

  11. Simplified0.4

    \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216}}}{\pi \cdot t}} \cdot \frac{\sqrt{2 \cdot 2 + \left(\left(v \cdot \left(v \cdot -6\right)\right) \cdot \left(v \cdot \left(v \cdot -6\right)\right) - 2 \cdot \left(v \cdot \left(v \cdot -6\right)\right)\right)}}{1 - v \cdot v}\]

  12. Simplified0.4

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216}}}{\pi \cdot t} \cdot \color{blue}{\frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v}}\]
  13. Using strategy rm

  14. Applied associate-/r*_binary64_17270.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216}}}{\pi}}{t}} \cdot \frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v}\]
  15. Using strategy rm

  16. Applied add-exp-log_binary64_18210.1

    \[\leadsto \frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{e^{\log \left(\sqrt{8 + {v}^{6} \cdot -216}\right)}}}}{\pi}}{t} \cdot \frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v}\]

  17. Applied add-exp-log_binary64_18210.1

    \[\leadsto \frac{\frac{\frac{\color{blue}{e^{\log \left(1 - 5 \cdot \left(v \cdot v\right)\right)}}}{e^{\log \left(\sqrt{8 + {v}^{6} \cdot -216}\right)}}}{\pi}}{t} \cdot \frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v}\]

  18. Applied div-exp_binary64_18340.1

    \[\leadsto \frac{\frac{\color{blue}{e^{\log \left(1 - 5 \cdot \left(v \cdot v\right)\right) - \log \left(\sqrt{8 + {v}^{6} \cdot -216}\right)}}}{\pi}}{t} \cdot \frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v}\]

  19. Simplified0.1

    \[\leadsto \frac{\frac{e^{\color{blue}{\log \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216}}\right)}}}{\pi}}{t} \cdot \frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v}\]

  20. Simplified0.1

    \[\leadsto \color{blue}{\frac{\frac{e^{\log \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216}}\right)}}{\pi}}{t} \cdot \frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v}}\]

  21. Final simplification0.1

    \[\leadsto \frac{\frac{e^{\log \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\sqrt{8 + {v}^{6} \cdot -216}}\right)}}{\pi}}{t} \cdot \frac{\sqrt{4 + \left({v}^{4} \cdot 36 - \left(v \cdot v\right) \cdot -12\right)}}{1 - v \cdot v}\]

Reproduce

herbie shell --seed 2021014 
(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)))))