VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 98.3%

Time: 16.3s

Alternatives: 8

Speedup: 3.7×

• could not determine a ground truth

  1. F = -2.0369106582509825e-225
  2. l = -940095330252375.5

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: 76.1% 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: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 4000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\pi \cdot l\_m}{F}, \frac{-1}{F}, \pi \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(l\_m \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \end{array} \]

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) 4000000000000.0)
    (fma (/ (* PI l_m) F) (/ -1.0 F) (* PI l_m))
    (* (* l_m (sqrt (sqrt PI))) (sqrt (* PI (sqrt PI)))))))

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) <= 4000000000000.0) {
		tmp = fma(((((double) M_PI) * l_m) / F), (-1.0 / F), (((double) M_PI) * l_m));
	} else {
		tmp = (l_m * sqrt(sqrt(((double) M_PI)))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	}
	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) <= 4000000000000.0)
		tmp = fma(Float64(Float64(pi * l_m) / F), Float64(-1.0 / F), Float64(pi * l_m));
	else
		tmp = Float64(Float64(l_m * sqrt(sqrt(pi))) * sqrt(Float64(pi * sqrt(pi))));
	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], 4000000000000.0], N[(N[(N[(Pi * l$95$m), $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision] + N[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(l$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 4e12

   1. Initial program 90.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

         \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \ell + \left(\mathsf{neg}\left(\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)\right)\right) + \mathsf{PI}\left(\right) \cdot \ell} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{F \cdot F} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}\right)\right) + \mathsf{PI}\left(\right) \cdot \ell \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \frac{1}{F \cdot F}}\right)\right) + \mathsf{PI}\left(\right) \cdot \ell \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot \color{blue}{\frac{1}{F \cdot F}}\right)\right) + \mathsf{PI}\left(\right) \cdot \ell \]

      7. un-div-inv N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F \cdot F}}\right)\right) + \mathsf{PI}\left(\right) \cdot \ell \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{\mathsf{neg}\left(F \cdot F\right)}} + \mathsf{PI}\left(\right) \cdot \ell \]

      9. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot 1}}{\mathsf{neg}\left(F \cdot F\right)} + \mathsf{PI}\left(\right) \cdot \ell \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot 1}{\mathsf{neg}\left(\color{blue}{F \cdot F}\right)} + \mathsf{PI}\left(\right) \cdot \ell \]

      11. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right) \cdot 1}{\color{blue}{F \cdot \left(\mathsf{neg}\left(F\right)\right)}} + \mathsf{PI}\left(\right) \cdot \ell \]

      12. times-frac N/A

          \[\leadsto \color{blue}{\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F} \cdot \frac{1}{\mathsf{neg}\left(F\right)}} + \mathsf{PI}\left(\right) \cdot \ell \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{F}\right)\right)} + \mathsf{PI}\left(\right) \cdot \ell \]

      14. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan \left(\mathsf{PI}\left(\right) \cdot \ell\right)}{F}, \mathsf{neg}\left(\frac{1}{F}\right), \mathsf{PI}\left(\right) \cdot \ell\right)} \]

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\ell \cdot \mathsf{PI}\left(\right)}}{F}, \frac{-1}{F}, \mathsf{PI}\left(\right) \cdot \ell\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-PI.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 4e12 < (*.f64 (PI.f64) l)

   1. Initial program 68.7%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 99.6

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

      \[\leadsto \left(\ell \cdot \sqrt{\sqrt{\pi}}\right) \cdot \color{blue}{\sqrt{\pi \cdot \sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 4000000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{\pi \cdot \ell}{F}, \frac{-1}{F}, \pi \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \sqrt{\sqrt{\pi}}\right) \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \pi \cdot l\_m + \tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F \cdot F}\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+189}:\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{-283}:\\ \;\;\;\;\frac{\pi \cdot l\_m}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \end{array} \]

FPCore

l\_m = (fabs.f64 l)
l\_s = (copysign.f64 #s(literal 1 binary64) l)
(FPCore (l_s F l_m)
 :precision binary64
 (let* ((t_0 (+ (* PI l_m) (* (tan (* PI l_m)) (/ -1.0 (* F F))))))
   (*
    l_s
    (if (<= t_0 -5e+189)
      (* PI l_m)
      (if (<= t_0 -2e-283) (/ (* PI l_m) (* F (- 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) {
	double t_0 = (((double) M_PI) * l_m) + (tan((((double) M_PI) * l_m)) * (-1.0 / (F * F)));
	double tmp;
	if (t_0 <= -5e+189) {
		tmp = ((double) M_PI) * l_m;
	} else if (t_0 <= -2e-283) {
		tmp = (((double) M_PI) * l_m) / (F * -F);
	} else {
		tmp = ((double) M_PI) * l_m;
	}
	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 t_0 = (Math.PI * l_m) + (Math.tan((Math.PI * l_m)) * (-1.0 / (F * F)));
	double tmp;
	if (t_0 <= -5e+189) {
		tmp = Math.PI * l_m;
	} else if (t_0 <= -2e-283) {
		tmp = (Math.PI * l_m) / (F * -F);
	} else {
		tmp = Math.PI * l_m;
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = (math.pi * l_m) + (math.tan((math.pi * l_m)) * (-1.0 / (F * F)))
	tmp = 0
	if t_0 <= -5e+189:
		tmp = math.pi * l_m
	elif t_0 <= -2e-283:
		tmp = (math.pi * l_m) / (F * -F)
	else:
		tmp = math.pi * l_m
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = Float64(Float64(pi * l_m) + Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / Float64(F * F))))
	tmp = 0.0
	if (t_0 <= -5e+189)
		tmp = Float64(pi * l_m);
	elseif (t_0 <= -2e-283)
		tmp = Float64(Float64(pi * l_m) / Float64(F * Float64(-F)));
	else
		tmp = Float64(pi * l_m);
	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)
	t_0 = (pi * l_m) + (tan((pi * l_m)) * (-1.0 / (F * F)));
	tmp = 0.0;
	if (t_0 <= -5e+189)
		tmp = pi * l_m;
	elseif (t_0 <= -2e-283)
		tmp = (pi * l_m) / (F * -F);
	else
		tmp = pi * l_m;
	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_] := Block[{t$95$0 = N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(l$95$s * If[LessEqual[t$95$0, -5e+189], N[(Pi * l$95$m), $MachinePrecision], If[LessEqual[t$95$0, -2e-283], N[(N[(Pi * l$95$m), $MachinePrecision] / N[(F * (-F)), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -5.0000000000000004e189 or -1.99999999999999989e-283 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l))))

   1. Initial program 74.2%

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

   3. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \mathsf{PI}\left(\right)} \]

      2. lower-PI.f64 83.3

         \[\leadsto \ell \cdot \color{blue}{\pi} \]

   5. Applied rewrites 83.3%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]

   if -5.0000000000000004e189 < (-.f64 (*.f64 (PI.f64) l) (*.f64 (/.f64 #s(literal 1 binary64) (*.f64 F F)) (tan.f64 (*.f64 (PI.f64) l)))) < -1.99999999999999989e-283

   1. Initial program 89.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)\right)} \]

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)\right)}\right) \]

      5. sub-neg N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\mathsf{PI}\left(\right) - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right)} \]

      7. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} - \frac{\mathsf{PI}\left(\right)}{{F}^{2}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \color{blue}{\frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right) \]

      9. lower-PI.f64 N/A

         \[\leadsto \ell \cdot \left(\mathsf{PI}\left(\right) - \frac{\color{blue}{\mathsf{PI}\left(\right)}}{{F}^{2}}\right) \]

      10. unpow2 N/A

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

      11. lower-*.f64 88.2

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in F around 0

      \[\leadsto \ell \cdot \left(-1 \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{{F}^{2}}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 86.4%

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

   11. Taylor expanded in F around 0

       \[\leadsto -1 \cdot \color{blue}{\frac{\ell \cdot \mathsf{PI}\left(\right)}{{F}^{2}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 86.4%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -5 \cdot 10^{+189}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell + \tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F \cdot F} \leq -2 \cdot 10^{-283}:\\ \;\;\;\;\frac{\pi \cdot \ell}{F \cdot \left(-F\right)}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))