VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.1% → 99.0%

Time: 57.9s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. F = 2.3003228227436805e+103
  2. l = 1220979460038061.5

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: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot l\_m\right)\\ t_1 := \cos \left(\pi \cdot l\_m\right)\\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 5000000000000:\\ \;\;\;\;\pi \cdot l\_m + \frac{\tan \left(\pi \cdot l\_m\right) \cdot \frac{-1}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \mathsf{expm1}\left(\frac{-0.5 \cdot \frac{{t\_0}^{2}}{{F}^{2} \cdot {t\_1}^{2}} + \frac{t\_0}{t\_1}}{{F}^{2}}\right)\\ \end{array} \end{array} \end{array} \]

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 (sin (* PI l_m))) (t_1 (cos (* PI l_m))))
   (*
    l_s
    (if (<= (* PI l_m) 5000000000000.0)
      (+ (* PI l_m) (/ (* (tan (* PI l_m)) (/ -1.0 F)) F))
      (-
       (* PI l_m)
       (expm1
        (/
         (+
          (* -0.5 (/ (pow t_0 2.0) (* (pow F 2.0) (pow t_1 2.0))))
          (/ t_0 t_1))
         (pow F 2.0))))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double t_0 = sin((((double) M_PI) * l_m));
	double t_1 = cos((((double) M_PI) * l_m));
	double tmp;
	if ((((double) M_PI) * l_m) <= 5000000000000.0) {
		tmp = (((double) M_PI) * l_m) + ((tan((((double) M_PI) * l_m)) * (-1.0 / F)) / F);
	} else {
		tmp = (((double) M_PI) * l_m) - expm1((((-0.5 * (pow(t_0, 2.0) / (pow(F, 2.0) * pow(t_1, 2.0)))) + (t_0 / t_1)) / pow(F, 2.0)));
	}
	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 t_0 = Math.sin((Math.PI * l_m));
	double t_1 = Math.cos((Math.PI * l_m));
	double tmp;
	if ((Math.PI * l_m) <= 5000000000000.0) {
		tmp = (Math.PI * l_m) + ((Math.tan((Math.PI * l_m)) * (-1.0 / F)) / F);
	} else {
		tmp = (Math.PI * l_m) - Math.expm1((((-0.5 * (Math.pow(t_0, 2.0) / (Math.pow(F, 2.0) * Math.pow(t_1, 2.0)))) + (t_0 / t_1)) / Math.pow(F, 2.0)));
	}
	return l_s * tmp;
}

Python

l\_m = math.fabs(l)
l\_s = math.copysign(1.0, l)
def code(l_s, F, l_m):
	t_0 = math.sin((math.pi * l_m))
	t_1 = math.cos((math.pi * l_m))
	tmp = 0
	if (math.pi * l_m) <= 5000000000000.0:
		tmp = (math.pi * l_m) + ((math.tan((math.pi * l_m)) * (-1.0 / F)) / F)
	else:
		tmp = (math.pi * l_m) - math.expm1((((-0.5 * (math.pow(t_0, 2.0) / (math.pow(F, 2.0) * math.pow(t_1, 2.0)))) + (t_0 / t_1)) / math.pow(F, 2.0)))
	return l_s * tmp

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	t_0 = sin(Float64(pi * l_m))
	t_1 = cos(Float64(pi * l_m))
	tmp = 0.0
	if (Float64(pi * l_m) <= 5000000000000.0)
		tmp = Float64(Float64(pi * l_m) + Float64(Float64(tan(Float64(pi * l_m)) * Float64(-1.0 / F)) / F));
	else
		tmp = Float64(Float64(pi * l_m) - expm1(Float64(Float64(Float64(-0.5 * Float64((t_0 ^ 2.0) / Float64((F ^ 2.0) * (t_1 ^ 2.0)))) + Float64(t_0 / t_1)) / (F ^ 2.0))));
	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_] := Block[{t$95$0 = N[Sin[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]}, N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 5000000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] + N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(Exp[N[(N[(N[(-0.5 * N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(N[Power[F, 2.0], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision] / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 5e12

   1. Initial program 79.7%

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

      1. associate-/r* 79.7%

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

      2. metadata-eval 79.7%

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

      3. add-sqr-sqrt 36.8%

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

      4. sqrt-prod 67.3%

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

      5. sqrt-div 67.3%

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

      6. associate-*l/ 67.3%

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

      7. sqrt-div 67.3%

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

      8. metadata-eval 67.3%

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

      9. sqrt-prod 40.5%

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

      10. add-sqr-sqrt 87.2%

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

   4. Applied egg-rr 87.2%

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

   if 5e12 < (*.f64 (PI.f64) l)

   1. Initial program 66.1%

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

      1. associate-*l/ 66.1%

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

      2. *-un-lft-identity 66.1%

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

      3. expm1-log1p-u 58.4%

         \[\leadsto \pi \cdot \ell - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\right)\right)} \]

      4. div-inv 58.4%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}}\right)\right) \]

      5. pow2 58.4%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{\color{blue}{{F}^{2}}}\right)\right) \]

      6. pow-flip 58.4%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(\pi \cdot \ell\right) \cdot \color{blue}{{F}^{\left(-2\right)}}\right)\right) \]

      7. metadata-eval 58.4%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(\pi \cdot \ell\right) \cdot {F}^{\color{blue}{-2}}\right)\right) \]

   4. Applied egg-rr 58.4%

      \[\leadsto \pi \cdot \ell - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(\pi \cdot \ell\right) \cdot {F}^{-2}\right)\right)} \]

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 5000000000000:\\ \;\;\;\;\pi \cdot \ell + \frac{\tan \left(\pi \cdot \ell\right) \cdot \frac{-1}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \mathsf{expm1}\left(\frac{-0.5 \cdot \frac{{\sin \left(\pi \cdot \ell\right)}^{2}}{{F}^{2} \cdot {\cos \left(\pi \cdot \ell\right)}^{2}} + \frac{\sin \left(\pi \cdot \ell\right)}{\cos \left(\pi \cdot \ell\right)}}{{F}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \left(\pi \cdot l\_m + \frac{1}{F \cdot \frac{F \cdot \left({l\_m}^{2} \cdot \left({\pi}^{3} \cdot \frac{0.3333333333333333}{{\pi}^{2}}\right)\right) - \frac{F}{\pi}}{l\_m}}\right) \end{array} \]

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
     (/
      (-
       (*
        F
        (* (pow l_m 2.0) (* (pow PI 3.0) (/ 0.3333333333333333 (pow PI 2.0)))))
       (/ 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 * (((F * (pow(l_m, 2.0) * (pow(((double) M_PI), 3.0) * (0.3333333333333333 / pow(((double) M_PI), 2.0))))) - (F / ((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.pow(l_m, 2.0) * (Math.pow(Math.PI, 3.0) * (0.3333333333333333 / Math.pow(Math.PI, 2.0))))) - (F / 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.pow(l_m, 2.0) * (math.pow(math.pi, 3.0) * (0.3333333333333333 / math.pow(math.pi, 2.0))))) - (F / 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(1.0 / Float64(F * Float64(Float64(Float64(F * Float64((l_m ^ 2.0) * Float64((pi ^ 3.0) * Float64(0.3333333333333333 / (pi ^ 2.0))))) - Float64(F / 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 * ((l_m ^ 2.0) * ((pi ^ 3.0) * (0.3333333333333333 / (pi ^ 2.0))))) - (F / 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[(1.0 / N[(F * N[(N[(N[(F * N[(N[Power[l$95$m, 2.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.3333333333333333 / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(F / Pi), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. associate-/r/ 76.8%

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

   2. pow2 76.8%

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

4. Applied egg-rr 76.8%

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

   1. pow2 76.8%

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

   2. associate-/l* 82.5%

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

6. Applied egg-rr 82.5%

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

7. Taylor expanded in l around 0 92.2%

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

9. Simplified 92.2%

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

10. Final simplification 92.2%

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

Alternative 3: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 10^{-57}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{l\_m}{F}}{\frac{F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m - \frac{\tan \left(\pi \cdot l\_m\right)}{F \cdot F}\\ \end{array} \end{array} \]

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) 1e-57)
    (- (* PI l_m) (/ (/ l_m F) (/ F PI)))
    (- (* 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) <= 1e-57) {
		tmp = (((double) M_PI) * l_m) - ((l_m / F) / (F / ((double) M_PI)));
	} 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) <= 1e-57) {
		tmp = (Math.PI * l_m) - ((l_m / F) / (F / Math.PI));
	} 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) <= 1e-57:
		tmp = (math.pi * l_m) - ((l_m / F) / (F / math.pi))
	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) <= 1e-57)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) / Float64(F / pi)));
	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) <= 1e-57)
		tmp = (pi * l_m) - ((l_m / F) / (F / pi));
	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], 1e-57], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] / N[(F / Pi), $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) < 9.99999999999999955e-58

   1. Initial program 79.0%

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

      1. *-commutative 79.0%

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

      2. sqr-neg 79.0%

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

      3. associate-*r/ 79.2%

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

      4. sqr-neg 79.2%

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

      5. *-rgt-identity 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in l around 0 74.3%

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

      1. *-commutative 74.3%

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

      2. times-frac 82.0%

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

   7. Applied egg-rr 82.0%

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

      1. *-commutative 82.0%

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

      2. clear-num 82.0%

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

      3. un-div-inv 82.0%

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

   9. Applied egg-rr 82.0%

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

   if 9.99999999999999955e-58 < (*.f64 (PI.f64) l)

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. sqr-neg 70.0%

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

      3. associate-*r/ 70.0%

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

      4. sqr-neg 70.0%

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

      5. *-rgt-identity 70.0%

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

   3. Simplified 70.0%

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

Alternative 4: 82.2% accurate, 1.0× speedup

Math

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

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)) (/ -1.0 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)) * (-1.0 / 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)) * (-1.0 / 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)) * (-1.0 / 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)) * Float64(-1.0 / 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)) * (-1.0 / 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] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. associate-/r* 76.7%

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

   2. metadata-eval 76.7%

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

   3. add-sqr-sqrt 35.2%

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

   4. sqrt-prod 66.9%

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

   5. sqrt-div 66.9%

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

   6. associate-*l/ 66.9%

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

   7. sqrt-div 66.9%

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

   8. metadata-eval 66.9%

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

   9. sqrt-prod 38.1%

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

   10. add-sqr-sqrt 82.5%

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

4. Applied egg-rr 82.5%

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

5. Final simplification 82.5%

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

Alternative 5: 82.2% accurate, 1.0× speedup

Math

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

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) (/ -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(-1.0 / Float64(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[(-1.0 / N[(F * N[(F / N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. associate-/r/ 76.8%

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

   2. pow2 76.8%

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

4. Applied egg-rr 76.8%

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

   1. pow2 76.8%

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

   2. associate-/l* 82.5%

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

6. Applied egg-rr 82.5%

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

7. Final simplification 82.5%

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

Alternative 6: 82.2% 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 76.7%

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

   1. associate-*l/ 76.8%

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

   2. *-un-lft-identity 76.8%

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

   3. associate-/r* 82.5%

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

4. Applied egg-rr 82.5%

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

Alternative 7: 74.9% accurate, 10.3× speedup

Math

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

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) (/ (/ l_m F) (/ F PI)))))

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) / (F / ((double) M_PI))));
}

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) / (F / Math.PI)));
}

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) / (F / math.pi)))

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(F / pi))))
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) / (F / pi)));
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[(F / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. *-commutative 76.7%

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

   2. sqr-neg 76.7%

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

   3. associate-*r/ 76.8%

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

   4. sqr-neg 76.8%

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

   5. *-rgt-identity 76.8%

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

3. Simplified 76.8%

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

5. Taylor expanded in l around 0 69.8%

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

   1. *-commutative 69.8%

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

   2. times-frac 75.6%

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

7. Applied egg-rr 75.6%

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

   1. *-commutative 75.6%

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

   2. clear-num 75.6%

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

   3. un-div-inv 75.6%

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

9. Applied egg-rr 75.6%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\ell}{F}}{\frac{F}{\pi}}} \]
10. Add Preprocessing

Alternative 8: 74.9% accurate, 10.3× speedup

Math

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

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) (/ (* l_m (/ PI 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) - ((l_m * (((double) M_PI) / 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) - ((l_m * (Math.PI / 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) - ((l_m * (math.pi / 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(l_m * Float64(pi / 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) - ((l_m * (pi / 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[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. associate-/r* 76.7%

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

   2. metadata-eval 76.7%

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

   3. add-sqr-sqrt 35.2%

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

   4. sqrt-prod 66.9%

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

   5. sqrt-div 66.9%

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

   6. associate-*l/ 66.9%

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

   7. sqrt-div 66.9%

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

   8. metadata-eval 66.9%

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

   9. sqrt-prod 38.1%

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

   10. add-sqr-sqrt 82.5%

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

4. Applied egg-rr 82.5%

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

5. Taylor expanded in l around 0 75.6%

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

   1. associate-/l* 75.6%

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

7. Simplified 75.6%

   \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\ell \cdot \frac{\pi}{F}}}{F} \]
8. Add Preprocessing

Alternative 9: 74.9% 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 #s(literal 1 binary64) 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 76.7%

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

   1. *-commutative 76.7%

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

   2. sqr-neg 76.7%

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

   3. associate-*r/ 76.8%

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

   4. sqr-neg 76.8%

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

   5. *-rgt-identity 76.8%

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

3. Simplified 76.8%

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

5. Taylor expanded in l around 0 69.8%

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

   1. *-commutative 69.8%

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

   2. times-frac 75.6%

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

7. Applied egg-rr 75.6%

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

8. Final simplification 75.6%

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

Reproduce

herbie shell --seed 2024090 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))