VandenBroeck and Keller, Equation (6)

Percentage Accurate: 77.0% → 98.7%

Time: 1.6min

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. F = -6.9474717492964e-96
  2. l = 1488986958186428.5

Specification

Math

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

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\log \left(e^{\frac{{\pi}^{2}}{l_m \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)}}\right)}, \frac{F}{\pi \cdot l_m}\right)}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (-
   (* PI l_m)
   (/
    (/ 1.0 F)
    (fma
     -1.0
     (/
      F
      (log (exp (/ (pow PI 2.0) (* l_m (* (pow PI 3.0) 0.3333333333333333))))))
     (/ F (* PI l_m)))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((1.0 / F) / fma(-1.0, (F / log(exp((pow(((double) M_PI), 2.0) / (l_m * (pow(((double) M_PI), 3.0) * 0.3333333333333333)))))), (F / (((double) M_PI) * l_m)))));
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(1.0 / F) / fma(-1.0, Float64(F / log(exp(Float64((pi ^ 2.0) / Float64(l_m * Float64((pi ^ 3.0) * 0.3333333333333333)))))), Float64(F / Float64(pi * l_m))))))
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(1.0 / F), $MachinePrecision] / N[(-1.0 * N[(F / N[Log[N[Exp[N[(N[Power[Pi, 2.0], $MachinePrecision] / N[(l$95$m * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(F / N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r/ 76.7%

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

   2. associate-/l* 81.0%

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

   3. clear-num 81.0%

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

   4. add-sqr-sqrt 38.9%

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

   5. sqrt-prod 66.1%

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

   6. sqr-neg 66.1%

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

   7. sqrt-unprod 28.6%

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

   8. add-sqr-sqrt 55.0%

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

   9. div-inv 55.0%

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

   10. clear-num 55.0%

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

   11. associate-*l/ 55.0%

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

   12. *-un-lft-identity 55.0%

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

   13. add-sqr-sqrt 28.6%

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

   14. sqrt-unprod 66.1%

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

   15. sqr-neg 66.1%

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

   16. sqrt-prod 38.9%

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

   17. add-sqr-sqrt 81.0%

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

4. Applied egg-rr 81.0%

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

5. Taylor expanded in l around 0 87.3%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{-1 \cdot \frac{F \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}} + \frac{F}{\ell \cdot \pi}}} \]
6. Step-by-step derivation

   1. fma-def 87.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\mathsf{fma}\left(-1, \frac{F \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}}, \frac{F}{\ell \cdot \pi}\right)}} \]

   2. associate-/l* 87.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \color{blue}{\frac{F}{\frac{{\pi}^{2}}{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}}}, \frac{F}{\ell \cdot \pi}\right)} \]

   3. distribute-rgt-out-- 87.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\frac{{\pi}^{2}}{\ell \cdot \color{blue}{\left({\pi}^{3} \cdot \left(-0.16666666666666666 - -0.5\right)\right)}}}, \frac{F}{\ell \cdot \pi}\right)} \]

   4. metadata-eval 87.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\frac{{\pi}^{2}}{\ell \cdot \left({\pi}^{3} \cdot \color{blue}{0.3333333333333333}\right)}}, \frac{F}{\ell \cdot \pi}\right)} \]

7. Simplified 87.3%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\mathsf{fma}\left(-1, \frac{F}{\frac{{\pi}^{2}}{\ell \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)}}, \frac{F}{\ell \cdot \pi}\right)}} \]
8. Step-by-step derivation

   1. add-log-exp 98.1%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\color{blue}{\log \left(e^{\frac{{\pi}^{2}}{\ell \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)}}\right)}}, \frac{F}{\ell \cdot \pi}\right)} \]

9. Applied egg-rr 98.1%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\color{blue}{\log \left(e^{\frac{{\pi}^{2}}{\ell \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)}}\right)}}, \frac{F}{\ell \cdot \pi}\right)} \]

10. Final simplification 98.1%

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\log \left(e^{\frac{{\pi}^{2}}{\ell \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)}}\right)}, \frac{F}{\pi \cdot \ell}\right)} \]
11. Add Preprocessing

Alternative 2: 88.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m + \frac{\frac{-1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\frac{3}{\pi \cdot l_m}}, \frac{F}{\pi \cdot l_m}\right)}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/ (/ -1.0 F) (fma -1.0 (/ F (/ 3.0 (* PI l_m))) (/ F (* PI l_m)))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + ((-1.0 / F) / fma(-1.0, (F / (3.0 / (((double) M_PI) * l_m))), (F / (((double) M_PI) * l_m)))));
}

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(Float64(-1.0 / F) / fma(-1.0, Float64(F / Float64(3.0 / Float64(pi * l_m))), Float64(F / Float64(pi * l_m))))))
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(-1.0 * N[(F / N[(3.0 / N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r/ 76.7%

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

   2. associate-/l* 81.0%

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

   3. clear-num 81.0%

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

   4. add-sqr-sqrt 38.9%

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

   5. sqrt-prod 66.1%

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

   6. sqr-neg 66.1%

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

   7. sqrt-unprod 28.6%

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

   8. add-sqr-sqrt 55.0%

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

   9. div-inv 55.0%

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

   10. clear-num 55.0%

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

   11. associate-*l/ 55.0%

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

   12. *-un-lft-identity 55.0%

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

   13. add-sqr-sqrt 28.6%

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

   14. sqrt-unprod 66.1%

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

   15. sqr-neg 66.1%

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

   16. sqrt-prod 38.9%

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

   17. add-sqr-sqrt 81.0%

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

4. Applied egg-rr 81.0%

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

5. Taylor expanded in l around 0 87.3%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{-1 \cdot \frac{F \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}} + \frac{F}{\ell \cdot \pi}}} \]
6. Step-by-step derivation

   1. fma-def 87.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\mathsf{fma}\left(-1, \frac{F \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}}, \frac{F}{\ell \cdot \pi}\right)}} \]

   2. associate-/l* 87.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \color{blue}{\frac{F}{\frac{{\pi}^{2}}{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}}}, \frac{F}{\ell \cdot \pi}\right)} \]

   3. distribute-rgt-out-- 87.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\frac{{\pi}^{2}}{\ell \cdot \color{blue}{\left({\pi}^{3} \cdot \left(-0.16666666666666666 - -0.5\right)\right)}}}, \frac{F}{\ell \cdot \pi}\right)} \]

   4. metadata-eval 87.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\frac{{\pi}^{2}}{\ell \cdot \left({\pi}^{3} \cdot \color{blue}{0.3333333333333333}\right)}}, \frac{F}{\ell \cdot \pi}\right)} \]

7. Simplified 87.3%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\mathsf{fma}\left(-1, \frac{F}{\frac{{\pi}^{2}}{\ell \cdot \left({\pi}^{3} \cdot 0.3333333333333333\right)}}, \frac{F}{\ell \cdot \pi}\right)}} \]

8. Taylor expanded in l around 0 87.3%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\color{blue}{\frac{3}{\ell \cdot \pi}}}, \frac{F}{\ell \cdot \pi}\right)} \]

9. Final simplification 87.3%

   \[\leadsto \pi \cdot \ell + \frac{\frac{-1}{F}}{\mathsf{fma}\left(-1, \frac{F}{\frac{3}{\pi \cdot \ell}}, \frac{F}{\pi \cdot \ell}\right)} \]
10. Add Preprocessing

Alternative 3: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\pi \cdot l_m - \frac{\pi \cdot \frac{l_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \frac{\tan \left(\pi \cdot l_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 5e-20)
    (- (* PI l_m) (/ (* PI (/ l_m F)) F))
    (- (* PI l_m) (/ (tan (* PI l_m)) (* F F))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 5e-20) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) * (l_m / F)) / F);
	} else {
		tmp = (((double) M_PI) * l_m) - (tan((((double) M_PI) * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 5e-20) {
		tmp = (Math.PI * l_m) - ((Math.PI * (l_m / F)) / F);
	} else {
		tmp = (Math.PI * l_m) - (Math.tan((Math.PI * l_m)) / (F * F));
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 5e-20:
		tmp = (math.pi * l_m) - ((math.pi * (l_m / F)) / F)
	else:
		tmp = (math.pi * l_m) - (math.tan((math.pi * l_m)) / (F * F))
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 5e-20)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(tan(Float64(pi * l_m)) / Float64(F * F)));
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 5e-20)
		tmp = (pi * l_m) - ((pi * (l_m / F)) / F);
	else
		tmp = (pi * l_m) - (tan((pi * l_m)) / (F * F));
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5e-20], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4.9999999999999999e-20

   1. Initial program 77.9%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 77.9%

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

      2. associate-*l/ 77.9%

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

      3. *-lft-identity 77.9%

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

      4. sqr-neg 77.9%

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

   3. Simplified 77.9%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 73.7%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
   6. Step-by-step derivation

      1. *-commutative 73.7%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

      2. times-frac 79.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   7. Applied egg-rr 79.7%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]
   8. Step-by-step derivation

      1. associate-*l/ 79.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi \cdot \frac{\ell}{F}}{F}} \]

   9. Applied egg-rr 79.7%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi \cdot \frac{\ell}{F}}{F}} \]

   if 4.9999999999999999e-20 < (*.f64 (PI.f64) l)

   1. Initial program 73.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 73.4%

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

      2. associate-*l/ 73.4%

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

      3. *-lft-identity 73.4%

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

      4. sqr-neg 73.4%

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

   3. Simplified 73.4%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot l_m\right)}}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (+ (* PI l_m) (/ (/ -1.0 F) (/ F (tan (* PI l_m)))))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) + ((-1.0 / F) / (F / tan((((double) M_PI) * l_m)))));
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) + ((-1.0 / F) / (F / Math.tan((Math.PI * l_m)))));
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) + ((-1.0 / F) / (F / math.tan((math.pi * l_m)))))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) + Float64(Float64(-1.0 / F) / Float64(F / tan(Float64(pi * l_m))))))
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) + ((-1.0 / F) / (F / tan((pi * l_m)))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F / N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r/ 76.7%

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

   2. associate-/l* 81.0%

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

   3. clear-num 81.0%

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

   4. add-sqr-sqrt 38.9%

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

   5. sqrt-prod 66.1%

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

   6. sqr-neg 66.1%

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

   7. sqrt-unprod 28.6%

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

   8. add-sqr-sqrt 55.0%

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

   9. div-inv 55.0%

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

   10. clear-num 55.0%

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

   11. associate-*l/ 55.0%

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

   12. *-un-lft-identity 55.0%

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

   13. add-sqr-sqrt 28.6%

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

   14. sqrt-unprod 66.1%

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

   15. sqr-neg 66.1%

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

   16. sqrt-prod 38.9%

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

   17. add-sqr-sqrt 81.0%

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

4. Applied egg-rr 81.0%

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

5. Final simplification 81.0%

   \[\leadsto \pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}} \]
6. Add Preprocessing

Alternative 5: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m - \frac{\frac{\tan \left(\pi \cdot l_m\right)}{F}}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (/ (tan (* PI l_m)) F) F))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((tan((((double) M_PI) * l_m)) / F) / F));
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((Math.tan((Math.PI * l_m)) / F) / F));
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((math.tan((math.pi * l_m)) / F) / F))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / F)))
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((tan((pi * l_m)) / F) / F));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 76.7%

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

   2. *-un-lft-identity 76.7%

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

   3. associate-/r* 81.0%

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

4. Applied egg-rr 81.0%

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

5. Final simplification 81.0%

   \[\leadsto \pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F} \]
6. Add Preprocessing

Alternative 6: 76.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l_m \leq 50000000000000:\\ \;\;\;\;\pi \cdot l_m - \frac{\pi \cdot \frac{l_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l_m - \frac{\frac{\frac{l_m}{\frac{F}{\pi}}}{-1}}{F}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 50000000000000.0)
    (- (* PI l_m) (/ (* PI (/ l_m F)) F))
    (- (* PI l_m) (/ (/ (/ l_m (/ F PI)) -1.0) F)))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((((double) M_PI) * l_m) <= 50000000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) * (l_m / F)) / F);
	} else {
		tmp = (((double) M_PI) * l_m) - (((l_m / (F / ((double) M_PI))) / -1.0) / F);
	}
	return l_s * tmp;
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	double tmp;
	if ((Math.PI * l_m) <= 50000000000000.0) {
		tmp = (Math.PI * l_m) - ((Math.PI * (l_m / F)) / F);
	} else {
		tmp = (Math.PI * l_m) - (((l_m / (F / Math.PI)) / -1.0) / F);
	}
	return l_s * tmp;
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	tmp = 0
	if (math.pi * l_m) <= 50000000000000.0:
		tmp = (math.pi * l_m) - ((math.pi * (l_m / F)) / F)
	else:
		tmp = (math.pi * l_m) - (((l_m / (F / math.pi)) / -1.0) / F)
	return l_s * tmp

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	tmp = 0.0
	if (Float64(pi * l_m) <= 50000000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F));
	else
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(Float64(l_m / Float64(F / pi)) / -1.0) / F));
	end
	return Float64(l_s * tmp)
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp_2 = code(l_s, F, l_m)
	tmp = 0.0;
	if ((pi * l_m) <= 50000000000000.0)
		tmp = (pi * l_m) - ((pi * (l_m / F)) / F);
	else
		tmp = (pi * l_m) - (((l_m / (F / pi)) / -1.0) / F);
	end
	tmp_2 = l_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 50000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[(l$95$m / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e13

   1. Initial program 78.6%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Step-by-step derivation

      1. sqr-neg 78.6%

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

      2. associate-*l/ 78.7%

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

      3. *-lft-identity 78.7%

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

      4. sqr-neg 78.7%

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

   3. Simplified 78.7%

      \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
   4. Add Preprocessing

   5. Taylor expanded in l around 0 74.4%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
   6. Step-by-step derivation

      1. *-commutative 74.4%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

      2. times-frac 80.1%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   7. Applied egg-rr 80.1%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]
   8. Step-by-step derivation

      1. associate-*l/ 80.1%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi \cdot \frac{\ell}{F}}{F}} \]

   9. Applied egg-rr 80.1%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi \cdot \frac{\ell}{F}}{F}} \]

   if 5e13 < (*.f64 (PI.f64) l)

   1. Initial program 70.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r/ 70.4%

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

      2. associate-/l* 70.4%

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

      3. clear-num 70.4%

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

      4. add-sqr-sqrt 27.8%

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

      5. sqrt-prod 68.3%

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

      6. sqr-neg 68.3%

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

      7. sqrt-unprod 40.5%

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

      8. add-sqr-sqrt 67.9%

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

      9. neg-mul-1 67.9%

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

      10. associate-/r* 67.9%

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

   4. Applied egg-rr 67.9%

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

   5. Taylor expanded in l around 0 53.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\color{blue}{\frac{\ell \cdot \pi}{F}}}{-1}}{F} \]
   6. Step-by-step derivation

      1. associate-/l* 53.3%

         \[\leadsto \pi \cdot \ell - \frac{\frac{\color{blue}{\frac{\ell}{\frac{F}{\pi}}}}{-1}}{F} \]

   7. Simplified 53.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\color{blue}{\frac{\ell}{\frac{F}{\pi}}}}{-1}}{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 50000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\frac{\ell}{\frac{F}{\pi}}}{-1}}{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m - \frac{l_m}{F} \cdot \frac{\pi}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (* (/ l_m F) (/ PI F)))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((l_m / F) * (((double) M_PI) / F)));
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((l_m / F) * (Math.PI / F)));
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((l_m / F) * (math.pi / F)))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) * Float64(pi / F))))
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((l_m / F) * (pi / F)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. sqr-neg 76.6%

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

   2. associate-*l/ 76.7%

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

   3. *-lft-identity 76.7%

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

   4. sqr-neg 76.7%

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

3. Simplified 76.7%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing

5. Taylor expanded in l around 0 68.3%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
6. Step-by-step derivation

   1. *-commutative 68.3%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

   2. times-frac 72.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

7. Applied egg-rr 72.6%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

8. Final simplification 72.6%

   \[\leadsto \pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F} \]
9. Add Preprocessing

Alternative 8: 75.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ l_s = \mathsf{copysign}\left(1, \ell\right) \\ l_s \cdot \left(\pi \cdot l_m - \frac{\pi \cdot \frac{l_m}{F}}{F}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
l_s = (copysign.f64 1 l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (/ (* PI (/ l_m F)) F))))

C

l_m = fabs(l);
l_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	return l_s * ((((double) M_PI) * l_m) - ((((double) M_PI) * (l_m / F)) / F));
}

Java

l_m = Math.abs(l);
l_s = Math.copySign(1.0, l);
public static double code(double l_s, double F, double l_m) {
	return l_s * ((Math.PI * l_m) - ((Math.PI * (l_m / F)) / F));
}

Python

l_m = math.fabs(l)
l_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	return l_s * ((math.pi * l_m) - ((math.pi * (l_m / F)) / F))

Julia

l_m = abs(l)
l_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / F)))
end

MATLAB

l_m = abs(l);
l_s = sign(l) * abs(1.0);
function tmp = code(l_s, F, l_m)
	tmp = l_s * ((pi * l_m) - ((pi * (l_m / F)) / F));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
l_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[l]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[l$95$s_, F_, l$95$m_] := N[(l$95$s * N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
2. Step-by-step derivation

   1. sqr-neg 76.6%

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

   2. associate-*l/ 76.7%

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

   3. *-lft-identity 76.7%

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

   4. sqr-neg 76.7%

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

3. Simplified 76.7%

   \[\leadsto \color{blue}{\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
4. Add Preprocessing

5. Taylor expanded in l around 0 68.3%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \pi}}{F \cdot F} \]
6. Step-by-step derivation

   1. *-commutative 68.3%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{F \cdot F} \]

   2. times-frac 72.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

7. Applied egg-rr 72.6%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]
8. Step-by-step derivation

   1. associate-*l/ 72.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi \cdot \frac{\ell}{F}}{F}} \]

9. Applied egg-rr 72.6%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi \cdot \frac{\ell}{F}}{F}} \]

10. Final simplification 72.6%

    \[\leadsto \pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))