Average Error: 17.4 → 8.8
Time: 1.1min
Precision: binary64
Cost: 2690
\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1.7804560111184658 \cdot 10^{+162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2.428756404616689 \cdot 10^{+151}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(\ell \cdot \sqrt{\pi}\right)\right)}}{F}\\ \end{array}\]
\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)
↓
\begin{array}{l}
\mathbf{if}\;\pi \cdot \ell \leq -1.7804560111184658 \cdot 10^{+162}:\\
\;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot F}}{F}\\

\mathbf{elif}\;\pi \cdot \ell \leq 2.428756404616689 \cdot 10^{+151}:\\
\;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)\right)}}{F}\\

\mathbf{else}:\\
\;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(\ell \cdot \sqrt{\pi}\right)\right)}}{F}\\

\end{array}
(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))
↓
(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -1.7804560111184658e+162)
   (-
    (* PI l)
    (/
     (/
      (sin (* PI l))
      (* (cos (* (cbrt (* PI l)) (* (cbrt (* PI l)) (cbrt (* PI l))))) F))
     F))
   (if (<= (* PI l) 2.428756404616689e+151)
     (-
      (* PI l)
      (/
       (/
        (sin (* PI l))
        (*
         F
         (-
          (+ 1.0 (* 0.041666666666666664 (pow (* PI l) 4.0)))
          (* (pow PI 2.0) (* (* l l) 0.5)))))
       F))
     (-
      (* PI l)
      (/
       (/
        (sin (* PI l))
        (*
         F
         (cos
          (*
           (* (cbrt (sqrt PI)) (* (cbrt (sqrt PI)) (cbrt (sqrt PI))))
           (* l (sqrt PI))))))
       F)))))
double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan(((double) M_PI) * l));
}
↓
double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= -1.7804560111184658e+162) {
		tmp = (((double) M_PI) * l) - ((sin(((double) M_PI) * l) / (cos(cbrt(((double) M_PI) * l) * (cbrt(((double) M_PI) * l) * cbrt(((double) M_PI) * l))) * F)) / F);
	} else if ((((double) M_PI) * l) <= 2.428756404616689e+151) {
		tmp = (((double) M_PI) * l) - ((sin(((double) M_PI) * l) / (F * ((1.0 + (0.041666666666666664 * pow((((double) M_PI) * l), 4.0))) - (pow(((double) M_PI), 2.0) * ((l * l) * 0.5))))) / F);
	} else {
		tmp = (((double) M_PI) * l) - ((sin(((double) M_PI) * l) / (F * cos((cbrt(sqrt((double) M_PI)) * (cbrt(sqrt((double) M_PI)) * cbrt(sqrt((double) M_PI)))) * (l * sqrt((double) M_PI))))) / F);
	}
	return tmp;
}

Error

Bits error versus F

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs
Alternative 1
Accuracy8.8
Cost2690
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1.7804560111184658 \cdot 10^{+162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2.428756404616689 \cdot 10^{+151}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\sqrt[3]{\ell} \cdot \left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right)\right)}{F}}{F}\\ \end{array}\]
Alternative 2
Accuracy8.8
Cost2690
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1.7804560111184658 \cdot 10^{+162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\ell \cdot \sqrt[3]{\pi}\right)\right) \cdot F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2.428756404616689 \cdot 10^{+151}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\sqrt[3]{\ell} \cdot \left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right)\right)}{F}}{F}\\ \end{array}\]
Alternative 3
Accuracy8.8
Cost2690
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1.7804560111184658 \cdot 10^{+162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt{\pi} \cdot \left(\ell \cdot \sqrt{\pi}\right)\right) \cdot F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2.428756404616689 \cdot 10^{+151}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\sqrt[3]{\ell} \cdot \left(\pi \cdot \left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right)\right)\right)}{F}}{F}\\ \end{array}\]
Alternative 4
Accuracy8.8
Cost2690
\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1.7804560111184658 \cdot 10^{+162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt{\pi} \cdot \left(\ell \cdot \sqrt{\pi}\right)\right) \cdot F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2.428756404616689 \cdot 10^{+151}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\sqrt{\ell} \cdot \left(\pi \cdot \sqrt{\ell}\right)\right)}{F}}{F}\\ \end{array}\]
Alternative 5
Accuracy12.8
Cost1344
\[\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt{\pi} \cdot \left(\ell \cdot \sqrt{\pi}\right)\right) \cdot F}}{F}\]
Alternative 6
Accuracy13.1
Cost1920
\[\pi \cdot \ell - \frac{\frac{\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \left(\sqrt[3]{\sin \left(\pi \cdot \ell\right)} \cdot \sqrt[3]{\sin \left(\pi \cdot \ell\right)}\right)}{\cos \left(\pi \cdot \ell\right) \cdot F}}{F}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 PI.f64 l) < -1.78045601111846582e162

    1. Initial program 22.6

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

    2. Simplified22.6

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\]
    3. Using strategy rm

    4. Applied associate-/r*_binary6422.6

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
    5. Using strategy rm

    6. Applied tan-quot_binary6422.6

      \[\leadsto \pi \cdot \ell - \frac{\frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F}}{F}\]

    7. Applied associate-/l/_binary6422.6

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}}}{F}\]

    8. Simplified22.6

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{\cos \left(\pi \cdot \ell\right) \cdot F}}}{F}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt_binary6422.5

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \color{blue}{\left(\left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right) \cdot \sqrt[3]{\pi \cdot \ell}\right)} \cdot F}}{F}\]

    if -1.78045601111846582e162 < (*.f64 PI.f64 l) < 2.42875640461669e151

    1. Initial program 15.9

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

    2. Simplified15.5

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\]
    3. Using strategy rm

    4. Applied associate-/r*_binary649.7

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
    5. Using strategy rm

    6. Applied tan-quot_binary649.7

      \[\leadsto \pi \cdot \ell - \frac{\frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F}}{F}\]

    7. Applied associate-/l/_binary649.7

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}}}{F}\]

    8. Simplified9.7

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{\cos \left(\pi \cdot \ell\right) \cdot F}}}{F}\]

    9. Taylor expanded around 0 4.2

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{\left(\left(0.041666666666666664 \cdot \left({\pi}^{4} \cdot {\ell}^{4}\right) + 1\right) - 0.5 \cdot \left({\pi}^{2} \cdot {\ell}^{2}\right)\right)} \cdot F}}{F}\]

    10. Simplified4.2

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{\left(\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)\right)} \cdot F}}{F}\]

    if 2.42875640461669e151 < (*.f64 PI.f64 l)

    1. Initial program 20.7

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right)\]

    2. Simplified20.7

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}}\]
    3. Using strategy rm

    4. Applied associate-/r*_binary6420.7

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}}\]
    5. Using strategy rm

    6. Applied tan-quot_binary6420.7

      \[\leadsto \pi \cdot \ell - \frac{\frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}}{F}}{F}\]

    7. Applied associate-/l/_binary6420.7

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\pi \cdot \ell\right)}}}{F}\]

    8. Simplified20.7

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\color{blue}{\cos \left(\pi \cdot \ell\right) \cdot F}}}{F}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt_binary6420.7

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \cdot \ell\right) \cdot F}}{F}\]

    11. Applied associate-*l*_binary6420.7

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \color{blue}{\left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \ell\right)\right)} \cdot F}}{F}\]

    12. Simplified20.7

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt{\pi} \cdot \color{blue}{\left(\ell \cdot \sqrt{\pi}\right)}\right) \cdot F}}{F}\]
    13. Using strategy rm

    14. Applied add-cube-cbrt_binary6420.6

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}\right)} \cdot \left(\ell \cdot \sqrt{\pi}\right)\right) \cdot F}}{F}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1.7804560111184658 \cdot 10^{+162}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\sqrt[3]{\pi \cdot \ell} \cdot \left(\sqrt[3]{\pi \cdot \ell} \cdot \sqrt[3]{\pi \cdot \ell}\right)\right) \cdot F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2.428756404616689 \cdot 10^{+151}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \left(\left(1 + 0.041666666666666664 \cdot {\left(\pi \cdot \ell\right)}^{4}\right) - {\pi}^{2} \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.5\right)\right)}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F \cdot \cos \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right) \cdot \left(\ell \cdot \sqrt{\pi}\right)\right)}}{F}\\ \end{array}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))