VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 99.0%

Time: 21.1s

Alternatives: 7

Speedup: 7.1×

• could not determine a ground truth

  1. F = 1.575858629009806e-278
  2. l = -1711144887707968.5

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: 76.6% 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, 1.0× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 100000000000:\\ \;\;\;\;\pi \cdot l\_m - \frac{\frac{\tan \left(\pi \cdot l\_m\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 100000000000.0)
    (- (* PI l_m) (/ (/ (tan (* 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 tmp;
	if ((((double) M_PI) * l_m) <= 100000000000.0) {
		tmp = (((double) M_PI) * l_m) - ((tan((((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 tmp;
	if ((Math.PI * l_m) <= 100000000000.0) {
		tmp = (Math.PI * l_m) - ((Math.tan((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):
	tmp = 0
	if (math.pi * l_m) <= 100000000000.0:
		tmp = (math.pi * l_m) - ((math.tan((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)
	tmp = 0.0
	if (Float64(pi * l_m) <= 100000000000.0)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(tan(Float64(pi * l_m)) / F) / 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)
	tmp = 0.0;
	if ((pi * l_m) <= 100000000000.0)
		tmp = (pi * l_m) - ((tan((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_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 100000000000.0], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l$95$m), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 1e11

   1. Initial program 81.8%

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

      1. associate-*l/ 82.1%

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

      2. *-un-lft-identity 82.1%

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

      3. associate-/r* 89.2%

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

   4. Applied egg-rr 89.2%

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

   if 1e11 < (*.f64 (PI.f64) l)

   1. Initial program 76.8%

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

   3. Taylor expanded in l around inf 99.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 100000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 2.2 \cdot 10^{-212} \lor \neg \left(F \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{l\_m \cdot \frac{\pi}{F}}{-F}\\ \end{array} \end{array} \]

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 (or (<= F 2.2e-212) (not (<= F 5e-12)))
    (* 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) {
	double tmp;
	if ((F <= 2.2e-212) || !(F <= 5e-12)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = (l_m * (((double) M_PI) / 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 ((F <= 2.2e-212) || !(F <= 5e-12)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = (l_m * (Math.PI / 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 (F <= 2.2e-212) or not (F <= 5e-12):
		tmp = math.pi * l_m
	else:
		tmp = (l_m * (math.pi / 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 ((F <= 2.2e-212) || !(F <= 5e-12))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(Float64(l_m * Float64(pi / F)) / Float64(-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 ((F <= 2.2e-212) || ~((F <= 5e-12)))
		tmp = pi * l_m;
	else
		tmp = (l_m * (pi / 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[Or[LessEqual[F, 2.2e-212], N[Not[LessEqual[F, 5e-12]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(N[(l$95$m * N[(Pi / F), $MachinePrecision]), $MachinePrecision] / (-F)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if F < 2.20000000000000003e-212 or 4.9999999999999997e-12 < F

   1. Initial program 84.1%

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

   3. Taylor expanded in l around inf 80.2%

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

   if 2.20000000000000003e-212 < F < 4.9999999999999997e-12

   1. Initial program 56.4%

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

   3. Taylor expanded in l around 0 46.1%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]

   4. Taylor expanded in F around 0 46.1%

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

      1. mul-1-neg 46.1%

         \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]

      2. associate-/l* 46.1%

         \[\leadsto -\color{blue}{\ell \cdot \frac{\pi}{{F}^{2}}} \]

      3. distribute-rgt-neg-in 46.1%

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

      4. distribute-frac-neg 46.1%

         \[\leadsto \ell \cdot \color{blue}{\frac{-\pi}{{F}^{2}}} \]

   6. Simplified 46.1%

      \[\leadsto \color{blue}{\ell \cdot \frac{-\pi}{{F}^{2}}} \]
   7. Step-by-step derivation

      1. pow2 46.1%

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

      2. clear-num 45.8%

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

      3. associate-*r/ 45.8%

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

      4. associate-/l* 45.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 2.9%

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

      7. sqr-neg 2.9%

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

      8. sqrt-unprod 2.9%

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

      9. add-sqr-sqrt 2.9%

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

      10. frac-times 3.0%

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

      11. clear-num 3.0%

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

      12. frac-2neg 3.0%

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

      13. associate-*r/ 3.0%

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

   8. Applied egg-rr 60.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.2 \cdot 10^{-212} \lor \neg \left(F \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\pi}{F}}{-F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 3 \cdot 10^{-213} \lor \neg \left(F \leq 3.1 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F \cdot \left(-\frac{F}{l\_m}\right)}\\ \end{array} \end{array} \]

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 (or (<= F 3e-213) (not (<= F 3.1e-12)))
    (* PI l_m)
    (/ PI (* F (- (/ F l_m)))))))

C

l\_m = fabs(l);
l\_s = copysign(1.0, l);
double code(double l_s, double F, double l_m) {
	double tmp;
	if ((F <= 3e-213) || !(F <= 3.1e-12)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = ((double) M_PI) / (F * -(F / 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 tmp;
	if ((F <= 3e-213) || !(F <= 3.1e-12)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = Math.PI / (F * -(F / 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):
	tmp = 0
	if (F <= 3e-213) or not (F <= 3.1e-12):
		tmp = math.pi * l_m
	else:
		tmp = math.pi / (F * -(F / l_m))
	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 ((F <= 3e-213) || !(F <= 3.1e-12))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(pi / Float64(F * Float64(-Float64(F / 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)
	tmp = 0.0;
	if ((F <= 3e-213) || ~((F <= 3.1e-12)))
		tmp = pi * l_m;
	else
		tmp = pi / (F * -(F / 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_] := N[(l$95$s * If[Or[LessEqual[F, 3e-213], N[Not[LessEqual[F, 3.1e-12]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(Pi / N[(F * (-N[(F / l$95$m), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if F < 2.99999999999999986e-213 or 3.1000000000000001e-12 < F

   1. Initial program 84.1%

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

   3. Taylor expanded in l around inf 80.2%

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

   if 2.99999999999999986e-213 < F < 3.1000000000000001e-12

   1. Initial program 56.4%

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

   3. Taylor expanded in l around 0 46.1%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]

   4. Taylor expanded in F around 0 46.1%

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

      1. mul-1-neg 46.1%

         \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]

      2. associate-/l* 46.1%

         \[\leadsto -\color{blue}{\ell \cdot \frac{\pi}{{F}^{2}}} \]

      3. distribute-rgt-neg-in 46.1%

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

      4. distribute-frac-neg 46.1%

         \[\leadsto \ell \cdot \color{blue}{\frac{-\pi}{{F}^{2}}} \]

   6. Simplified 46.1%

      \[\leadsto \color{blue}{\ell \cdot \frac{-\pi}{{F}^{2}}} \]
   7. Step-by-step derivation

      1. pow2 46.1%

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

      2. clear-num 45.8%

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

      3. associate-*r/ 45.8%

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

      4. associate-/l* 45.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 2.9%

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

      7. sqr-neg 2.9%

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

      8. sqrt-unprod 2.9%

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

      9. add-sqr-sqrt 2.9%

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

      10. frac-times 3.0%

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

      11. clear-num 3.0%

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

      12. frac-2neg 3.0%

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

      13. clear-num 3.0%

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

      14. frac-times 3.0%

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

   8. Applied egg-rr 60.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3 \cdot 10^{-213} \lor \neg \left(F \leq 3.1 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F \cdot \left(-\frac{F}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 6.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{-203} \lor \neg \left(F \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot l\_m\\ \mathbf{else}:\\ \;\;\;\;l\_m \cdot \frac{\frac{\pi}{F}}{-F}\\ \end{array} \end{array} \]

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 (or (<= F 1.25e-203) (not (<= F 4e-12)))
    (* 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) {
	double tmp;
	if ((F <= 1.25e-203) || !(F <= 4e-12)) {
		tmp = ((double) M_PI) * l_m;
	} else {
		tmp = l_m * ((((double) M_PI) / 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 ((F <= 1.25e-203) || !(F <= 4e-12)) {
		tmp = Math.PI * l_m;
	} else {
		tmp = l_m * ((Math.PI / 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 (F <= 1.25e-203) or not (F <= 4e-12):
		tmp = math.pi * l_m
	else:
		tmp = l_m * ((math.pi / 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 ((F <= 1.25e-203) || !(F <= 4e-12))
		tmp = Float64(pi * l_m);
	else
		tmp = Float64(l_m * Float64(Float64(pi / F) / Float64(-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 ((F <= 1.25e-203) || ~((F <= 4e-12)))
		tmp = pi * l_m;
	else
		tmp = l_m * ((pi / 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[Or[LessEqual[F, 1.25e-203], N[Not[LessEqual[F, 4e-12]], $MachinePrecision]], N[(Pi * l$95$m), $MachinePrecision], N[(l$95$m * N[(N[(Pi / F), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if F < 1.25e-203 or 3.99999999999999992e-12 < F

   1. Initial program 83.1%

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

   3. Taylor expanded in l around inf 80.0%

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

   if 1.25e-203 < F < 3.99999999999999992e-12

   1. Initial program 62.1%

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

   3. Taylor expanded in l around 0 50.8%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]

   4. Taylor expanded in F around 0 50.8%

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

      1. mul-1-neg 50.8%

         \[\leadsto \color{blue}{-\frac{\ell \cdot \pi}{{F}^{2}}} \]

      2. associate-/l* 50.7%

         \[\leadsto -\color{blue}{\ell \cdot \frac{\pi}{{F}^{2}}} \]

      3. distribute-rgt-neg-in 50.7%

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

      4. distribute-frac-neg 50.7%

         \[\leadsto \ell \cdot \color{blue}{\frac{-\pi}{{F}^{2}}} \]

   6. Simplified 50.7%

      \[\leadsto \color{blue}{\ell \cdot \frac{-\pi}{{F}^{2}}} \]
   7. Step-by-step derivation

      1. neg-mul-1 50.7%

         \[\leadsto \ell \cdot \frac{\color{blue}{-1 \cdot \pi}}{{F}^{2}} \]

      2. pow2 50.7%

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

      3. times-frac 50.7%

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

   8. Applied egg-rr 50.7%

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

      1. associate-*l/ 50.7%

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

      2. associate-*r/ 50.7%

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

      3. neg-mul-1 50.7%

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

   10. Applied egg-rr 50.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{-203} \lor \neg \left(F \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\pi}{F}}{-F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot l\_m \leq 0.002:\\ \;\;\;\;\pi \cdot l\_m - \frac{\pi \cdot \frac{l\_m}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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) 0.002) (- (* PI l_m) (/ (* 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 tmp;
	if ((((double) M_PI) * l_m) <= 0.002) {
		tmp = (((double) M_PI) * l_m) - ((((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 tmp;
	if ((Math.PI * l_m) <= 0.002) {
		tmp = (Math.PI * l_m) - ((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):
	tmp = 0
	if (math.pi * l_m) <= 0.002:
		tmp = (math.pi * l_m) - ((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)
	tmp = 0.0
	if (Float64(pi * l_m) <= 0.002)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(pi * Float64(l_m / F)) / 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)
	tmp = 0.0;
	if ((pi * l_m) <= 0.002)
		tmp = (pi * l_m) - ((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_] := N[(l$95$s * If[LessEqual[N[(Pi * l$95$m), $MachinePrecision], 0.002], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(Pi * N[(l$95$m / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2e-3

   1. Initial program 81.6%

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

      1. associate-*l/ 81.9%

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

      2. *-un-lft-identity 81.9%

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

      3. associate-/r* 89.0%

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

   4. Applied egg-rr 89.0%

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

      1. clear-num 89.0%

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

      2. inv-pow 89.0%

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

   6. Applied egg-rr 89.0%

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

      1. unpow-1 89.0%

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

      2. *-commutative 89.0%

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

   8. Simplified 89.0%

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

   9. Taylor expanded in l around 0 83.2%

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

       1. *-commutative 83.2%

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

       2. associate-/r* 83.1%

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

   11. Simplified 83.1%

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

       1. clear-num 83.2%

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

       2. associate-/r/ 83.2%

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

   13. Applied egg-rr 83.2%

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

   if 2e-3 < (*.f64 (PI.f64) l)

   1. Initial program 78.0%

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

   3. Taylor expanded in l around inf 98.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 0.002:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} l\_m = \left|\ell\right| \\ l\_s = \mathsf{copysign}\left(1, \ell\right) \\ l\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 0.5:\\ \;\;\;\;\pi \cdot l\_m - \frac{l\_m}{F} \cdot \frac{\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot l\_m\\ \end{array} \end{array} \]

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 (<= l_m 0.5) (- (* PI l_m) (* (/ l_m F) (/ PI 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 tmp;
	if (l_m <= 0.5) {
		tmp = (((double) M_PI) * l_m) - ((l_m / F) * (((double) M_PI) / 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 tmp;
	if (l_m <= 0.5) {
		tmp = (Math.PI * l_m) - ((l_m / F) * (Math.PI / 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):
	tmp = 0
	if l_m <= 0.5:
		tmp = (math.pi * l_m) - ((l_m / F) * (math.pi / 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)
	tmp = 0.0
	if (l_m <= 0.5)
		tmp = Float64(Float64(pi * l_m) - Float64(Float64(l_m / F) * Float64(pi / 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)
	tmp = 0.0;
	if (l_m <= 0.5)
		tmp = (pi * l_m) - ((l_m / F) * (pi / 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_] := N[(l$95$s * If[LessEqual[l$95$m, 0.5], N[(N[(Pi * l$95$m), $MachinePrecision] - N[(N[(l$95$m / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 0.5

   1. Initial program 81.6%

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

   3. Taylor expanded in l around 0 76.0%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative 76.0%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{{F}^{2}} \]

      2. pow2 76.0%

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

      3. times-frac 83.2%

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

   5. Applied egg-rr 83.2%

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

   if 0.5 < l

   1. Initial program 78.0%

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

   3. Taylor expanded in l around inf 98.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.5:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 37.7× speedup

Math

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

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)))

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);
}

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);
}

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)

Julia

l\_m = abs(l)
l\_s = copysign(1.0, l)
function code(l_s, F, l_m)
	return Float64(l_s * 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);
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[(Pi * l$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.8%

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

3. Taylor expanded in l around inf 75.3%

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

4. Final simplification 75.3%

   \[\leadsto \pi \cdot \ell \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))