VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.5% → 99.0%

Time: 13.5s

Alternatives: 8

Speedup: 0.8×

• could not determine a ground truth

  1. F = -8.34190248331624e-220
  2. l = 83944688499810.5

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: 76.5% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 200000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+17) (not (<= (* PI l) 200000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -2e+17) || !((((double) M_PI) * l) <= 200000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -2e+17) || !((Math.PI * l) <= 200000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -2e+17) or not ((math.pi * l) <= 200000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -2e+17) || !(Float64(pi * l) <= 200000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+17) || ~(((pi * l) <= 200000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+17], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 200000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e17 or 2e8 < (*.f64 (PI.f64) l)

   1. Initial program 63.9%

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

      1. sqr-neg 63.9%

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

      2. associate-*l/ 63.9%

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

      3. *-lft-identity 63.9%

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

      4. sqr-neg 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in l around 0 46.2%

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

      1. unpow2 46.2%

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

   6. Simplified 46.2%

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

   7. Taylor expanded in F around inf 99.6%

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

   if -2e17 < (*.f64 (PI.f64) l) < 2e8

   1. Initial program 86.7%

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

      1. associate-*l/ 89.2%

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

      2. *-un-lft-identity 89.2%

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

      3. associate-/r* 98.8%

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

   3. Applied egg-rr 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+17} \lor \neg \left(\pi \cdot \ell \leq 200000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2: 73.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{-60}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq -1 \cdot 10^{-181}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-248}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-214} \lor \neg \left(\pi \cdot \ell \leq 10^{-112}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (let* ((t_0 (* (/ PI F) (/ (- l) F))))
   (if (<= (* PI l) -5e-60)
     (* PI l)
     (if (<= (* PI l) -1e-181)
       t_0
       (if (<= (* PI l) 5e-248)
         (* PI l)
         (if (<= (* PI l) 5e-229)
           t_0
           (if (or (<= (* PI l) 5e-214) (not (<= (* PI l) 1e-112)))
             (* PI l)
             (* (/ (/ l F) F) (- PI)))))))))

C

double code(double F, double l) {
	double t_0 = (((double) M_PI) / F) * (-l / F);
	double tmp;
	if ((((double) M_PI) * l) <= -5e-60) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= -1e-181) {
		tmp = t_0;
	} else if ((((double) M_PI) * l) <= 5e-248) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= 5e-229) {
		tmp = t_0;
	} else if (((((double) M_PI) * l) <= 5e-214) || !((((double) M_PI) * l) <= 1e-112)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = ((l / F) / F) * -((double) M_PI);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double t_0 = (Math.PI / F) * (-l / F);
	double tmp;
	if ((Math.PI * l) <= -5e-60) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= -1e-181) {
		tmp = t_0;
	} else if ((Math.PI * l) <= 5e-248) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= 5e-229) {
		tmp = t_0;
	} else if (((Math.PI * l) <= 5e-214) || !((Math.PI * l) <= 1e-112)) {
		tmp = Math.PI * l;
	} else {
		tmp = ((l / F) / F) * -Math.PI;
	}
	return tmp;
}

Python

def code(F, l):
	t_0 = (math.pi / F) * (-l / F)
	tmp = 0
	if (math.pi * l) <= -5e-60:
		tmp = math.pi * l
	elif (math.pi * l) <= -1e-181:
		tmp = t_0
	elif (math.pi * l) <= 5e-248:
		tmp = math.pi * l
	elif (math.pi * l) <= 5e-229:
		tmp = t_0
	elif ((math.pi * l) <= 5e-214) or not ((math.pi * l) <= 1e-112):
		tmp = math.pi * l
	else:
		tmp = ((l / F) / F) * -math.pi
	return tmp

Julia

function code(F, l)
	t_0 = Float64(Float64(pi / F) * Float64(Float64(-l) / F))
	tmp = 0.0
	if (Float64(pi * l) <= -5e-60)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= -1e-181)
		tmp = t_0;
	elseif (Float64(pi * l) <= 5e-248)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= 5e-229)
		tmp = t_0;
	elseif ((Float64(pi * l) <= 5e-214) || !(Float64(pi * l) <= 1e-112))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(Float64(l / F) / F) * Float64(-pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	t_0 = (pi / F) * (-l / F);
	tmp = 0.0;
	if ((pi * l) <= -5e-60)
		tmp = pi * l;
	elseif ((pi * l) <= -1e-181)
		tmp = t_0;
	elseif ((pi * l) <= 5e-248)
		tmp = pi * l;
	elseif ((pi * l) <= 5e-229)
		tmp = t_0;
	elseif (((pi * l) <= 5e-214) || ~(((pi * l) <= 1e-112)))
		tmp = pi * l;
	else
		tmp = ((l / F) / F) * -pi;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := Block[{t$95$0 = N[(N[(Pi / F), $MachinePrecision] * N[((-l) / F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(Pi * l), $MachinePrecision], -5e-60], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], -1e-181], t$95$0, If[LessEqual[N[(Pi * l), $MachinePrecision], 5e-248], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 5e-229], t$95$0, If[Or[LessEqual[N[(Pi * l), $MachinePrecision], 5e-214], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e-112]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(N[(l / F), $MachinePrecision] / F), $MachinePrecision] * (-Pi)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < -5.0000000000000001e-60 or -1.00000000000000005e-181 < (*.f64 (PI.f64) l) < 5.0000000000000001e-248 or 5.00000000000000016e-229 < (*.f64 (PI.f64) l) < 4.9999999999999998e-214 or 9.9999999999999995e-113 < (*.f64 (PI.f64) l)

   1. Initial program 73.9%

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

      1. sqr-neg 73.9%

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

      2. associate-*l/ 74.2%

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

      3. *-lft-identity 74.2%

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

      4. sqr-neg 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in l around 0 60.7%

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

      1. unpow2 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in F around inf 90.2%

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

   if -5.0000000000000001e-60 < (*.f64 (PI.f64) l) < -1.00000000000000005e-181 or 5.0000000000000001e-248 < (*.f64 (PI.f64) l) < 5.00000000000000016e-229

   1. Initial program 73.9%

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

      1. sqr-neg 73.9%

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

      2. associate-*l/ 80.8%

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

      3. *-lft-identity 80.8%

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

      4. sqr-neg 80.8%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in l around 0 74.1%

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

      1. unpow2 74.1%

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

   6. Simplified 74.1%

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

      1. associate-/r* 74.1%

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

      2. div-inv 74.1%

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

   8. Applied egg-rr 74.1%

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

   9. Taylor expanded in F around 0 54.7%

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

       1. mul-1-neg 54.7%

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

       2. unpow2 54.7%

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

   11. Simplified 54.7%

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

       1. *-commutative 54.7%

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

       2. times-frac 73.5%

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

   13. Applied egg-rr 73.5%

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

   if 4.9999999999999998e-214 < (*.f64 (PI.f64) l) < 9.9999999999999995e-113

   1. Initial program 95.0%

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

      1. sqr-neg 95.0%

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

      2. associate-*l/ 95.0%

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

      3. *-lft-identity 95.0%

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

      4. sqr-neg 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in l around 0 94.9%

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

      1. unpow2 94.9%

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

   6. Simplified 94.9%

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

      1. associate-/r* 94.9%

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

      2. div-inv 95.0%

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

   8. Applied egg-rr 95.0%

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

   9. Taylor expanded in F around 0 61.9%

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

       1. mul-1-neg 61.9%

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

       2. unpow2 61.9%

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

   11. Simplified 61.9%

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

       1. associate-/r* 66.6%

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

       2. associate-*l/ 66.7%

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

       3. associate-/l* 66.6%

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

       4. associate-/r/ 66.6%

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

   13. Applied egg-rr 66.6%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{-60}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq -1 \cdot 10^{-181}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-248}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-229}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 5 \cdot 10^{-214} \lor \neg \left(\pi \cdot \ell \leq 10^{-112}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{F}}{F} \cdot \left(-\pi\right)\\ \end{array} \]

Alternative 3: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -50000000000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -50000000000.0) (not (<= (* PI l) 5e-5)))
   (* PI l)
   (- (* PI l) (* PI (/ l (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -50000000000.0) || !((((double) M_PI) * l) <= 5e-5)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - (((double) M_PI) * (l / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -50000000000.0) || !((Math.PI * l) <= 5e-5)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - (Math.PI * (l / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -50000000000.0) or not ((math.pi * l) <= 5e-5):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - (math.pi * (l / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -50000000000.0) || !(Float64(pi * l) <= 5e-5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(pi * Float64(l / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -50000000000.0) || ~(((pi * l) <= 5e-5)))
		tmp = pi * l;
	else
		tmp = (pi * l) - (pi * (l / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -50000000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5e-5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(Pi * N[(l / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e10 or 5.00000000000000024e-5 < (*.f64 (PI.f64) l)

   1. Initial program 65.0%

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

      1. sqr-neg 65.0%

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

      2. associate-*l/ 65.0%

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

      3. *-lft-identity 65.0%

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

      4. sqr-neg 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in l around 0 46.0%

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

      1. unpow2 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in F around inf 96.8%

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

   if -5e10 < (*.f64 (PI.f64) l) < 5.00000000000000024e-5

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0 88.9%

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

      1. associate-/l* 88.6%

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

      2. associate-/r/ 88.9%

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

      3. unpow2 88.9%

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

   4. Simplified 88.9%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -50000000000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 4: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -50000000000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -50000000000.0) (not (<= (* PI l) 5e-5)))
   (* PI l)
   (- (* PI l) (* (/ l F) (/ PI F)))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -50000000000.0) || !((((double) M_PI) * l) <= 5e-5)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((l / F) * (((double) M_PI) / F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -50000000000.0) || !((Math.PI * l) <= 5e-5)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / F) * (Math.PI / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -50000000000.0) or not ((math.pi * l) <= 5e-5):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((l / F) * (math.pi / F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -50000000000.0) || !(Float64(pi * l) <= 5e-5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / F) * Float64(pi / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -50000000000.0) || ~(((pi * l) <= 5e-5)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / F) * (pi / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -50000000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5e-5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / F), $MachinePrecision] * N[(Pi / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e10 or 5.00000000000000024e-5 < (*.f64 (PI.f64) l)

   1. Initial program 65.0%

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

      1. sqr-neg 65.0%

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

      2. associate-*l/ 65.0%

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

      3. *-lft-identity 65.0%

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

      4. sqr-neg 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in l around 0 46.0%

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

      1. unpow2 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in F around inf 96.8%

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

   if -5e10 < (*.f64 (PI.f64) l) < 5.00000000000000024e-5

   1. Initial program 86.7%

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

      1. sqr-neg 86.7%

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

      2. associate-*l/ 89.5%

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

      3. *-lft-identity 89.5%

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

      4. sqr-neg 89.5%

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

   3. Simplified 89.5%

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

      1. add-cbrt-cube 64.9%

         \[\leadsto \pi \cdot \ell - \frac{\tan \color{blue}{\left(\sqrt[3]{\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right) \cdot \left(\pi \cdot \ell\right)}\right)}}{F \cdot F} \]

      2. pow1/3 45.7%

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

      3. pow3 45.7%

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

      4. pow-to-exp 28.6%

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

      5. pow-exp 43.1%

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

   5. Applied egg-rr 43.1%

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

   6. Taylor expanded in l around 0 88.9%

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

      1. *-commutative 88.9%

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

      2. unpow2 88.9%

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

      3. times-frac 99.0%

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

   8. Simplified 99.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -50000000000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \]

Alternative 5: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -50000000000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -50000000000.0) (not (<= (* PI l) 5e-5)))
   (* PI l)
   (- (* PI l) (/ (* PI (/ l F)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -50000000000.0) || !((((double) M_PI) * l) <= 5e-5)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) * (l / F)) / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -50000000000.0) || !((Math.PI * l) <= 5e-5)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.PI * (l / F)) / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -50000000000.0) or not ((math.pi * l) <= 5e-5):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.pi * (l / F)) / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -50000000000.0) || !(Float64(pi * l) <= 5e-5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi * Float64(l / F)) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -50000000000.0) || ~(((pi * l) <= 5e-5)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((pi * (l / F)) / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -50000000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5e-5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e10 or 5.00000000000000024e-5 < (*.f64 (PI.f64) l)

   1. Initial program 65.0%

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

      1. sqr-neg 65.0%

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

      2. associate-*l/ 65.0%

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

      3. *-lft-identity 65.0%

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

      4. sqr-neg 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in l around 0 46.0%

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

      1. unpow2 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in F around inf 96.8%

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

   if -5e10 < (*.f64 (PI.f64) l) < 5.00000000000000024e-5

   1. Initial program 86.7%

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

      1. associate-*l/ 89.5%

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

      2. *-un-lft-identity 89.5%

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

      3. associate-/r* 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in l around 0 99.0%

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

      1. associate-/l* 99.0%

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

      2. associate-/r/ 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -50000000000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \]

Alternative 6: 92.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -50000000000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -50000000000.0) (not (<= (* PI l) 5e-5)))
   (* PI l)
   (* l (- PI (/ PI (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -50000000000.0) || !((((double) M_PI) * l) <= 5e-5)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = l * (((double) M_PI) - (((double) M_PI) / (F * F)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (((Math.PI * l) <= -50000000000.0) || !((Math.PI * l) <= 5e-5)) {
		tmp = Math.PI * l;
	} else {
		tmp = l * (Math.PI - (Math.PI / (F * F)));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if ((math.pi * l) <= -50000000000.0) or not ((math.pi * l) <= 5e-5):
		tmp = math.pi * l
	else:
		tmp = l * (math.pi - (math.pi / (F * F)))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -50000000000.0) || !(Float64(pi * l) <= 5e-5))
		tmp = Float64(pi * l);
	else
		tmp = Float64(l * Float64(pi - Float64(pi / Float64(F * F))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -50000000000.0) || ~(((pi * l) <= 5e-5)))
		tmp = pi * l;
	else
		tmp = l * (pi - (pi / (F * F)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -50000000000.0], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5e-5]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(l * N[(Pi - N[(Pi / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e10 or 5.00000000000000024e-5 < (*.f64 (PI.f64) l)

   1. Initial program 65.0%

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

      1. sqr-neg 65.0%

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

      2. associate-*l/ 65.0%

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

      3. *-lft-identity 65.0%

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

      4. sqr-neg 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in l around 0 46.0%

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

      1. unpow2 46.0%

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

   6. Simplified 46.0%

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

   7. Taylor expanded in F around inf 96.8%

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

   if -5e10 < (*.f64 (PI.f64) l) < 5.00000000000000024e-5

   1. Initial program 86.7%

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

      1. sqr-neg 86.7%

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

      2. associate-*l/ 89.5%

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

      3. *-lft-identity 89.5%

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

      4. sqr-neg 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in l around 0 86.2%

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

      1. unpow2 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -50000000000 \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 7: 73.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \cdot 10^{-61} \lor \neg \left(\ell \leq -4 \cdot 10^{-182}\right) \land \left(\ell \leq 3 \cdot 10^{-248} \lor \neg \left(\ell \leq 3.8 \cdot 10^{-227} \lor \neg \left(\ell \leq 5.2 \cdot 10^{-214}\right) \land \ell \leq 3.6 \cdot 10^{-111}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -5.3e-61)
         (and (not (<= l -4e-182))
              (or (<= l 3e-248)
                  (not
                   (or (<= l 3.8e-227)
                       (and (not (<= l 5.2e-214)) (<= l 3.6e-111)))))))
   (* PI l)
   (* (/ PI F) (/ (- l) F))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -5.3e-61) || (!(l <= -4e-182) && ((l <= 3e-248) || !((l <= 3.8e-227) || (!(l <= 5.2e-214) && (l <= 3.6e-111)))))) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) / F) * (-l / F);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -5.3e-61) || (!(l <= -4e-182) && ((l <= 3e-248) || !((l <= 3.8e-227) || (!(l <= 5.2e-214) && (l <= 3.6e-111)))))) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI / F) * (-l / F);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -5.3e-61) or (not (l <= -4e-182) and ((l <= 3e-248) or not ((l <= 3.8e-227) or (not (l <= 5.2e-214) and (l <= 3.6e-111))))):
		tmp = math.pi * l
	else:
		tmp = (math.pi / F) * (-l / F)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -5.3e-61) || (!(l <= -4e-182) && ((l <= 3e-248) || !((l <= 3.8e-227) || (!(l <= 5.2e-214) && (l <= 3.6e-111))))))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi / F) * Float64(Float64(-l) / F));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -5.3e-61) || (~((l <= -4e-182)) && ((l <= 3e-248) || ~(((l <= 3.8e-227) || (~((l <= 5.2e-214)) && (l <= 3.6e-111)))))))
		tmp = pi * l;
	else
		tmp = (pi / F) * (-l / F);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -5.3e-61], And[N[Not[LessEqual[l, -4e-182]], $MachinePrecision], Or[LessEqual[l, 3e-248], N[Not[Or[LessEqual[l, 3.8e-227], And[N[Not[LessEqual[l, 5.2e-214]], $MachinePrecision], LessEqual[l, 3.6e-111]]]], $MachinePrecision]]]], N[(Pi * l), $MachinePrecision], N[(N[(Pi / F), $MachinePrecision] * N[((-l) / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.3e-61 or -4.0000000000000002e-182 < l < 3.00000000000000014e-248 or 3.8000000000000001e-227 < l < 5.2e-214 or 3.6000000000000001e-111 < l

   1. Initial program 73.9%

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

      1. sqr-neg 73.9%

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

      2. associate-*l/ 74.2%

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

      3. *-lft-identity 74.2%

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

      4. sqr-neg 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in l around 0 60.7%

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

      1. unpow2 60.7%

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

   6. Simplified 60.7%

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

   7. Taylor expanded in F around inf 90.2%

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

   if -5.3e-61 < l < -4.0000000000000002e-182 or 3.00000000000000014e-248 < l < 3.8000000000000001e-227 or 5.2e-214 < l < 3.6000000000000001e-111

   1. Initial program 81.0%

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

      1. sqr-neg 81.0%

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

      2. associate-*l/ 85.6%

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

      3. *-lft-identity 85.6%

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

      4. sqr-neg 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in l around 0 81.0%

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

      1. unpow2 81.0%

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

   6. Simplified 81.0%

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

      1. associate-/r* 81.0%

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

      2. div-inv 81.0%

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

   8. Applied egg-rr 81.0%

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

   9. Taylor expanded in F around 0 57.1%

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

       1. mul-1-neg 57.1%

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

       2. unpow2 57.1%

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

   11. Simplified 57.1%

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

       1. *-commutative 57.1%

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

       2. times-frac 71.1%

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

   13. Applied egg-rr 71.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \cdot 10^{-61} \lor \neg \left(\ell \leq -4 \cdot 10^{-182}\right) \land \left(\ell \leq 3 \cdot 10^{-248} \lor \neg \left(\ell \leq 3.8 \cdot 10^{-227} \lor \neg \left(\ell \leq 5.2 \cdot 10^{-214}\right) \land \ell \leq 3.6 \cdot 10^{-111}\right)\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \end{array} \]

Alternative 8: 73.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell \end{array} \]

FPCore

(FPCore (F l) :precision binary64 (* PI l))

C

double code(double F, double l) {
	return ((double) M_PI) * l;
}

Java

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

Python

def code(F, l):
	return math.pi * l

Julia

function code(F, l)
	return Float64(pi * l)
end

MATLAB

function tmp = code(F, l)
	tmp = pi * l;
end

Wolfram

code[F_, l_] := N[(Pi * l), $MachinePrecision]

Derivation

1. Initial program 75.5%

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

   1. sqr-neg 75.5%

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

   2. associate-*l/ 76.8%

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

   3. *-lft-identity 76.8%

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

   4. sqr-neg 76.8%

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

3. Simplified 76.8%

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

4. Taylor expanded in l around 0 65.5%

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

   1. unpow2 65.5%

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

6. Simplified 65.5%

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

7. Taylor expanded in F around inf 76.2%

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

8. Final simplification 76.2%

   \[\leadsto \pi \cdot \ell \]

Reproduce

herbie shell --seed 2023263 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))