VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.2% → 98.9%

Time: 37.2s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. F = 1.1464131868184945e-304
  2. l = 1440335657057852.5

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: 76.2% 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: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 500000000:\\ \;\;\;\;\pi \cdot l\_m + \frac{-1}{\frac{F}{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot l\_m\right)}{F \cdot \left(-1 - {l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {l\_m}^{2} \cdot \left(-0.001388888888888889 \cdot \left({l\_m}^{2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 500000000.0)
    (+ (* PI l_m) (/ -1.0 (/ F (/ (tan (* PI l_m)) F))))
    (+
     (* PI l_m)
     (/
      (* (/ 1.0 F) (sin (* PI l_m)))
      (*
       F
       (-
        -1.0
        (*
         (pow l_m 2.0)
         (+
          (* -0.5 (pow PI 2.0))
          (*
           (pow l_m 2.0)
           (+
            (*
             -0.001388888888888889
             (* (pow l_m 2.0) (log1p (expm1 (pow PI 6.0)))))
            (* 0.041666666666666664 (pow PI 4.0)))))))))))))

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) <= 500000000.0) {
		tmp = (((double) M_PI) * l_m) + (-1.0 / (F / (tan((((double) M_PI) * l_m)) / F)));
	} else {
		tmp = (((double) M_PI) * l_m) + (((1.0 / F) * sin((((double) M_PI) * l_m))) / (F * (-1.0 - (pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (pow(l_m, 2.0) * ((-0.001388888888888889 * (pow(l_m, 2.0) * log1p(expm1(pow(((double) M_PI), 6.0))))) + (0.041666666666666664 * pow(((double) M_PI), 4.0)))))))));
	}
	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) <= 500000000.0) {
		tmp = (Math.PI * l_m) + (-1.0 / (F / (Math.tan((Math.PI * l_m)) / F)));
	} else {
		tmp = (Math.PI * l_m) + (((1.0 / F) * Math.sin((Math.PI * l_m))) / (F * (-1.0 - (Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (Math.pow(l_m, 2.0) * ((-0.001388888888888889 * (Math.pow(l_m, 2.0) * Math.log1p(Math.expm1(Math.pow(Math.PI, 6.0))))) + (0.041666666666666664 * Math.pow(Math.PI, 4.0)))))))));
	}
	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) <= 500000000.0:
		tmp = (math.pi * l_m) + (-1.0 / (F / (math.tan((math.pi * l_m)) / F)))
	else:
		tmp = (math.pi * l_m) + (((1.0 / F) * math.sin((math.pi * l_m))) / (F * (-1.0 - (math.pow(l_m, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (math.pow(l_m, 2.0) * ((-0.001388888888888889 * (math.pow(l_m, 2.0) * math.log1p(math.expm1(math.pow(math.pi, 6.0))))) + (0.041666666666666664 * math.pow(math.pi, 4.0)))))))))
	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) <= 500000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(-1.0 / Float64(F / Float64(tan(Float64(pi * l_m)) / F))));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(Float64(1.0 / F) * sin(Float64(pi * l_m))) / Float64(F * Float64(-1.0 - Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64((l_m ^ 2.0) * Float64(Float64(-0.001388888888888889 * Float64((l_m ^ 2.0) * log1p(expm1((pi ^ 6.0))))) + Float64(0.041666666666666664 * (pi ^ 4.0))))))))));
	end
	return Float64(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], 500000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(-1.0 / N[(F / N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[(1.0 / F), $MachinePrecision] * N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(F * N[(-1.0 - N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.001388888888888889 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Log[1 + N[(Exp[N[Power[Pi, 6.0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e8

   1. Initial program 78.5%

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

      1. associate-*l/ 78.7%

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

      2. *-un-lft-identity 78.7%

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

      3. associate-/r* 87.1%

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

      4. clear-num 87.1%

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

   4. Applied egg-rr 87.1%

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

   if 5e8 < (*.f64 (PI.f64) l)

   1. Initial program 65.7%

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

      1. add-sqr-sqrt 65.7%

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

      2. sqrt-div 65.7%

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

      3. metadata-eval 65.7%

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

      4. sqrt-prod 31.5%

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

      5. add-sqr-sqrt 65.4%

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

      6. div-inv 65.4%

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

      7. tan-quot 65.4%

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

      8. frac-times 65.4%

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

      9. sqrt-div 65.4%

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

      10. metadata-eval 65.4%

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

      11. sqrt-prod 31.5%

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

      12. add-sqr-sqrt 65.7%

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

   4. Applied egg-rr 65.7%

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

   5. Taylor expanded in l around 0 96.5%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \color{blue}{\left(1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)\right)}} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 99.5%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)\right)} \]

   7. Applied egg-rr 99.5%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 500000000:\\ \;\;\;\;\pi \cdot \ell + \frac{-1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \left(-1 - {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\pi}^{6}\right)\right)\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{\sin \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F}}{F \cdot \left(1 + {l\_m}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {l\_m}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({l\_m}^{2} \cdot {\pi}^{6}\right)\right)\right)\right)}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/
    (* (sin (* PI l_m)) (/ -1.0 F))
    (*
     F
     (+
      1.0
      (*
       (pow l_m 2.0)
       (+
        (* -0.5 (pow PI 2.0))
        (*
         (pow l_m 2.0)
         (+
          (* 0.041666666666666664 (pow PI 4.0))
          (* -0.001388888888888889 (* (pow l_m 2.0) (pow PI 6.0)))))))))))))

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) + ((sin((((double) M_PI) * l_m)) * (-1.0 / F)) / (F * (1.0 + (pow(l_m, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (pow(l_m, 2.0) * ((0.041666666666666664 * pow(((double) M_PI), 4.0)) + (-0.001388888888888889 * (pow(l_m, 2.0) * pow(((double) M_PI), 6.0)))))))))));
}

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.sin((Math.PI * l_m)) * (-1.0 / F)) / (F * (1.0 + (Math.pow(l_m, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (Math.pow(l_m, 2.0) * ((0.041666666666666664 * Math.pow(Math.PI, 4.0)) + (-0.001388888888888889 * (Math.pow(l_m, 2.0) * Math.pow(Math.PI, 6.0)))))))))));
}

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.sin((math.pi * l_m)) * (-1.0 / F)) / (F * (1.0 + (math.pow(l_m, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (math.pow(l_m, 2.0) * ((0.041666666666666664 * math.pow(math.pi, 4.0)) + (-0.001388888888888889 * (math.pow(l_m, 2.0) * math.pow(math.pi, 6.0)))))))))))

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(sin(Float64(pi * l_m)) * Float64(-1.0 / F)) / Float64(F * Float64(1.0 + Float64((l_m ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64((l_m ^ 2.0) * Float64(Float64(0.041666666666666664 * (pi ^ 4.0)) + Float64(-0.001388888888888889 * Float64((l_m ^ 2.0) * (pi ^ 6.0))))))))))))
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) + ((sin((pi * l_m)) * (-1.0 / F)) / (F * (1.0 + ((l_m ^ 2.0) * ((-0.5 * (pi ^ 2.0)) + ((l_m ^ 2.0) * ((0.041666666666666664 * (pi ^ 4.0)) + (-0.001388888888888889 * ((l_m ^ 2.0) * (pi ^ 6.0)))))))))));
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[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / N[(F * N[(1.0 + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.001388888888888889 * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

   1. add-sqr-sqrt 75.2%

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

   2. sqrt-div 75.2%

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

   3. metadata-eval 75.2%

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

   4. sqrt-prod 36.8%

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

   5. add-sqr-sqrt 66.6%

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

   6. div-inv 66.6%

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

   7. tan-quot 66.6%

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

   8. frac-times 66.6%

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

   9. sqrt-div 66.8%

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

   10. metadata-eval 66.8%

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

   11. sqrt-prod 40.0%

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

   12. add-sqr-sqrt 81.7%

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

4. Applied egg-rr 81.7%

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

5. Taylor expanded in l around 0 97.3%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \color{blue}{\left(1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)\right)}} \]

6. Final simplification 97.3%

   \[\leadsto \pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F \cdot \left(1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(0.041666666666666664 \cdot {\pi}^{4} + -0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right)\right)\right)\right)} \]
7. Add Preprocessing

Alternative 3: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 100000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{1}{\frac{F}{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m + \frac{\sin \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F}}{-0.001388888888888889 \cdot \left({\pi}^{6} \cdot \left(F \cdot {l\_m}^{6}\right)\right)}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 100000000.0)
    (- (* PI l_m) (/ 1.0 (/ F (/ (tan (* PI l_m)) F))))
    (+
     (* PI l_m)
     (/
      (* (sin (* PI l_m)) (/ -1.0 F))
      (* -0.001388888888888889 (* (pow PI 6.0) (* F (pow l_m 6.0)))))))))

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) <= 100000000.0) {
		tmp = (((double) M_PI) * l_m) - (1.0 / (F / (tan((((double) M_PI) * l_m)) / F)));
	} else {
		tmp = (((double) M_PI) * l_m) + ((sin((((double) M_PI) * l_m)) * (-1.0 / F)) / (-0.001388888888888889 * (pow(((double) M_PI), 6.0) * (F * pow(l_m, 6.0)))));
	}
	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) <= 100000000.0) {
		tmp = (Math.PI * l_m) - (1.0 / (F / (Math.tan((Math.PI * l_m)) / F)));
	} else {
		tmp = (Math.PI * l_m) + ((Math.sin((Math.PI * l_m)) * (-1.0 / F)) / (-0.001388888888888889 * (Math.pow(Math.PI, 6.0) * (F * Math.pow(l_m, 6.0)))));
	}
	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) <= 100000000.0:
		tmp = (math.pi * l_m) - (1.0 / (F / (math.tan((math.pi * l_m)) / F)))
	else:
		tmp = (math.pi * l_m) + ((math.sin((math.pi * l_m)) * (-1.0 / F)) / (-0.001388888888888889 * (math.pow(math.pi, 6.0) * (F * math.pow(l_m, 6.0)))))
	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) <= 100000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(1.0 / Float64(F / Float64(tan(Float64(pi * l_m)) / F))));
	else
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(sin(Float64(pi * l_m)) * Float64(-1.0 / F)) / Float64(-0.001388888888888889 * Float64((pi ^ 6.0) * Float64(F * (l_m ^ 6.0))))));
	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) <= 100000000.0)
		tmp = (pi * l_m) - (1.0 / (F / (tan((pi * l_m)) / F)));
	else
		tmp = (pi * l_m) + ((sin((pi * l_m)) * (-1.0 / F)) / (-0.001388888888888889 * ((pi ^ 6.0) * (F * (l_m ^ 6.0)))));
	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], 100000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(1.0 / N[(F / N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / N[(-0.001388888888888889 * N[(N[Power[Pi, 6.0], $MachinePrecision] * N[(F * N[Power[l$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e8

   1. Initial program 78.4%

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

      1. associate-*l/ 78.6%

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

      2. *-un-lft-identity 78.6%

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

      3. associate-/r* 87.0%

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

      4. clear-num 87.1%

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

   4. Applied egg-rr 87.1%

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

   if 1e8 < (*.f64 (PI.f64) l)

   1. Initial program 66.2%

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

      1. add-sqr-sqrt 66.2%

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

      2. sqrt-div 66.2%

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

      3. metadata-eval 66.2%

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

      4. sqrt-prod 31.0%

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

      5. add-sqr-sqrt 64.4%

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

      6. div-inv 64.4%

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

      7. tan-quot 64.4%

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

      8. frac-times 64.4%

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

      9. sqrt-div 64.4%

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

      10. metadata-eval 64.4%

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

      11. sqrt-prod 31.0%

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

      12. add-sqr-sqrt 66.2%

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

   4. Applied egg-rr 66.2%

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

   5. Taylor expanded in l around 0 96.6%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{F \cdot \color{blue}{\left(1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)\right)}} \]

   6. Taylor expanded in l around inf 96.6%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{\color{blue}{-0.001388888888888889 \cdot \left(F \cdot \left({\ell}^{6} \cdot {\pi}^{6}\right)\right)}} \]
   7. Step-by-step derivation

      1. associate-*r* 96.6%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{-0.001388888888888889 \cdot \color{blue}{\left(\left(F \cdot {\ell}^{6}\right) \cdot {\pi}^{6}\right)}} \]

      2. *-commutative 96.6%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{-0.001388888888888889 \cdot \color{blue}{\left({\pi}^{6} \cdot \left(F \cdot {\ell}^{6}\right)\right)}} \]

   8. Simplified 96.6%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F} \cdot \sin \left(\pi \cdot \ell\right)}{\color{blue}{-0.001388888888888889 \cdot \left({\pi}^{6} \cdot \left(F \cdot {\ell}^{6}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 100000000:\\ \;\;\;\;\pi \cdot \ell - \frac{1}{\frac{F}{\frac{\tan \left(\pi \cdot \ell\right)}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{-0.001388888888888889 \cdot \left({\pi}^{6} \cdot \left(F \cdot {\ell}^{6}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{\sin \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F}}{F \cdot \mathsf{fma}\left(-0.5, {\left(\pi \cdot l\_m\right)}^{2}, 1\right)}\right) \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (+
   (* PI l_m)
   (/
    (* (sin (* PI l_m)) (/ -1.0 F))
    (* F (fma -0.5 (pow (* PI l_m) 2.0) 1.0))))))

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) + ((sin((((double) M_PI) * l_m)) * (-1.0 / F)) / (F * fma(-0.5, pow((((double) M_PI) * l_m), 2.0), 1.0))));
}

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(sin(Float64(pi * l_m)) * Float64(-1.0 / F)) / Float64(F * fma(-0.5, (Float64(pi * l_m) ^ 2.0), 1.0)))))
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[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / N[(F * N[(-0.5 * N[Power[N[(Pi * l$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

   1. add-sqr-sqrt 75.2%

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

   2. sqrt-div 75.2%

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

   3. metadata-eval 75.2%

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

   4. sqrt-prod 36.8%

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

   5. add-sqr-sqrt 66.6%

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

   6. div-inv 66.6%

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

   7. tan-quot 66.6%

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

   8. frac-times 66.6%

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

   9. sqrt-div 66.8%

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

   10. metadata-eval 66.8%

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

   11. sqrt-prod 40.0%

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

   12. add-sqr-sqrt 81.7%

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

4. Applied egg-rr 81.7%

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

5. Taylor expanded in l around 0 92.4%

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

   1. +-commutative 92.4%

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

   2. fma-define 92.4%

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

   3. *-commutative 92.4%

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

   4. unpow2 92.4%

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

   5. unpow2 92.4%

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

   6. swap-sqr 92.4%

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

   7. unpow2 92.4%

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

   8. *-commutative 92.4%

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

7. Simplified 92.4%

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

8. Final simplification 92.4%

   \[\leadsto \pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F \cdot \mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)} \]
9. Add Preprocessing

Alternative 5: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi}{F} \cdot \frac{l\_m}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\tan \left(\pi \cdot l\_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (*
  l_s
  (if (<= (* PI l_m) 2e-68)
    (- (* PI l_m) (* (/ PI F) (/ l_m 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) <= 2e-68) {
		tmp = (((double) M_PI) * l_m) - ((((double) M_PI) / F) * (l_m / 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) <= 2e-68) {
		tmp = (Math.PI * l_m) - ((Math.PI / F) * (l_m / 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) <= 2e-68:
		tmp = (math.pi * l_m) - ((math.pi / F) * (l_m / 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) <= 2e-68)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi / F) * Float64(l_m / 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) <= 2e-68)
		tmp = (pi * l_m) - ((pi / F) * (l_m / 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], 2e-68], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $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) < 2.00000000000000013e-68

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. sqr-neg 76.8%

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

      3. associate-*r/ 77.0%

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

      4. sqr-neg 77.0%

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

      5. *-rgt-identity 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in l around 0 70.9%

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

      1. *-commutative 70.9%

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

      2. times-frac 80.1%

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

   7. Applied egg-rr 80.1%

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

   if 2.00000000000000013e-68 < (*.f64 (PI.f64) l)

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

      2. sqr-neg 71.8%

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

      3. associate-*r/ 71.8%

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

      4. sqr-neg 71.8%

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

      5. *-rgt-identity 71.8%

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

   3. Simplified 71.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-68}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 1.0× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (+ (* PI l_m) (/ -1.0 (/ F (/ (tan (* PI l_m)) 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) + (-1.0 / (F / (tan((((double) M_PI) * l_m)) / 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) + (-1.0 / (F / (Math.tan((Math.PI * l_m)) / 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) + (-1.0 / (F / (math.tan((math.pi * l_m)) / 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(-1.0 / Float64(F / Float64(tan(Float64(pi * l_m)) / 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) + (-1.0 / (F / (tan((pi * l_m)) / 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[(-1.0 / N[(F / N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

   1. associate-*l/ 75.4%

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

   2. *-un-lft-identity 75.4%

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

   3. associate-/r* 81.7%

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

   4. clear-num 81.7%

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

4. Applied egg-rr 81.7%

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

5. Final simplification 81.7%

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

Alternative 7: 81.9% 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 #s(literal 1 binary64) 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 75.2%

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

   1. associate-*l/ 75.4%

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

   2. *-un-lft-identity 75.4%

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

   3. associate-/r* 81.7%

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

4. Applied egg-rr 81.7%

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

5. Final simplification 81.7%

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

Alternative 8: 74.2% accurate, 10.3× speedup

Math

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

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (* l_s (- (* PI l_m) (* (/ PI F) (/ l_m 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) / F) * (l_m / 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 / F) * (l_m / 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 / F) * (l_m / 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 / F) * Float64(l_m / 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 / F) * (l_m / 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 / F), $MachinePrecision] * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.2%

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

   1. *-commutative 75.2%

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

   2. sqr-neg 75.2%

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

   3. associate-*r/ 75.4%

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

   4. sqr-neg 75.4%

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

   5. *-rgt-identity 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in l around 0 67.3%

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

   1. *-commutative 67.3%

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

   2. times-frac 73.6%

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

7. Applied egg-rr 73.6%

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

8. Final simplification 73.6%

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

Reproduce

herbie shell --seed 2024078 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))