VandenBroeck and Keller, Equation (6)

Percentage Accurate: 77.0% → 97.2%

Time: 16.6s

Alternatives: 7

Speedup: 1.5×

• could not determine a ground truth

  1. F = 2.192035385762351e+138
  2. l = -3557522065159502.5

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: 77.0% 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: 97.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+36} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{-\mathsf{fma}\left({\ell}^{3}, \frac{{\pi}^{3}}{F} \cdot 0.3333333333333333, \pi \cdot \frac{\ell}{F}\right)}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+36) (not (<= (* PI l) 2e-6)))
   (* PI l)
   (fma
    PI
    l
    (/
     (-
      (fma
       (pow l 3.0)
       (* (/ (pow PI 3.0) F) 0.3333333333333333)
       (* PI (/ l F))))
     F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+36) || !((((double) M_PI) * l) <= 2e-6)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = fma(((double) M_PI), l, (-fma(pow(l, 3.0), ((pow(((double) M_PI), 3.0) / F) * 0.3333333333333333), (((double) M_PI) * (l / F))) / F));
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -5e+36) || !(Float64(pi * l) <= 2e-6))
		tmp = Float64(pi * l);
	else
		tmp = fma(pi, l, Float64(Float64(-fma((l ^ 3.0), Float64(Float64((pi ^ 3.0) / F) * 0.3333333333333333), Float64(pi * Float64(l / F)))) / F));
	end
	return tmp
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -5e+36], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e-6]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * l + N[((-N[(N[Power[l, 3.0], $MachinePrecision] * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] / F), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(Pi * N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -4.99999999999999977e36 or 1.99999999999999991e-6 < (*.f64 (PI.f64) l)

   1. Initial program 63.5%

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

      1. sqr-neg 63.5%

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

      2. associate-*l/ 63.5%

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

      3. *-lft-identity 63.5%

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

      4. sqr-neg 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in l around 0 48.7%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -4.99999999999999977e36 < (*.f64 (PI.f64) l) < 1.99999999999999991e-6

   1. Initial program 90.3%

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

      1. *-commutative 90.3%

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

      2. sqr-neg 90.3%

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

      3. *-commutative 90.3%

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

      4. fma-neg 90.3%

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

      5. associate-*l/ 90.3%

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

      6. times-frac 99.5%

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

      7. distribute-lft-neg-in 99.5%

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

      8. neg-mul-1 99.5%

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

      9. associate-/r* 99.5%

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

      10. metadata-eval 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. metadata-eval 99.5%

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

      13. times-frac 90.3%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in l around 0 99.6%

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

      1. distribute-lft-out 99.6%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{-1 \cdot \left({\ell}^{3} \cdot \left(-0.16666666666666666 \cdot \frac{{\pi}^{3}}{F} - -0.5 \cdot \frac{{\pi}^{3}}{F}\right) + \frac{\ell \cdot \pi}{F}\right)}}{F}\right) \]

      2. mul-1-neg 99.6%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{-\left({\ell}^{3} \cdot \left(-0.16666666666666666 \cdot \frac{{\pi}^{3}}{F} - -0.5 \cdot \frac{{\pi}^{3}}{F}\right) + \frac{\ell \cdot \pi}{F}\right)}}{F}\right) \]

      3. fma-def 99.6%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{-\color{blue}{\mathsf{fma}\left({\ell}^{3}, -0.16666666666666666 \cdot \frac{{\pi}^{3}}{F} - -0.5 \cdot \frac{{\pi}^{3}}{F}, \frac{\ell \cdot \pi}{F}\right)}}{F}\right) \]

      4. distribute-rgt-out-- 99.6%

         \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{-\mathsf{fma}\left({\ell}^{3}, \color{blue}{\frac{{\pi}^{3}}{F} \cdot \left(-0.16666666666666666 - -0.5\right)}, \frac{\ell \cdot \pi}{F}\right)}{F}\right) \]

      5. metadata-eval 99.6%

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

      6. associate-/l* 99.6%

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

      7. associate-/r/ 99.6%

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

   6. Simplified 99.6%

      \[\leadsto \mathsf{fma}\left(\pi, \ell, \frac{\color{blue}{-\mathsf{fma}\left({\ell}^{3}, \frac{{\pi}^{3}}{F} \cdot 0.3333333333333333, \frac{\ell}{F} \cdot \pi\right)}}{F}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+36} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{-\mathsf{fma}\left({\ell}^{3}, \frac{{\pi}^{3}}{F} \cdot 0.3333333333333333, \pi \cdot \frac{\ell}{F}\right)}{F}\right)\\ \end{array} \]

Alternative 2: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+36} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+36) (not (<= (* PI l) 2e-6)))
   (* PI l)
   (- (* PI l) (/ (* PI (/ l F)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -5e+36) || !((((double) M_PI) * l) <= 2e-6)) {
		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) <= -5e+36) || !((Math.PI * l) <= 2e-6)) {
		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) <= -5e+36) or not ((math.pi * l) <= 2e-6):
		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) <= -5e+36) || !(Float64(pi * l) <= 2e-6))
		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) <= -5e+36) || ~(((pi * l) <= 2e-6)))
		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], -5e+36], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e-6]], $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) < -4.99999999999999977e36 or 1.99999999999999991e-6 < (*.f64 (PI.f64) l)

   1. Initial program 63.5%

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

      1. sqr-neg 63.5%

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

      2. associate-*l/ 63.5%

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

      3. *-lft-identity 63.5%

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

      4. sqr-neg 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in l around 0 48.7%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -4.99999999999999977e36 < (*.f64 (PI.f64) l) < 1.99999999999999991e-6

   1. Initial program 90.3%

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

      1. sqr-neg 90.3%

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

      2. associate-*l/ 90.3%

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

      3. *-lft-identity 90.3%

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

      4. sqr-neg 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in l around 0 90.3%

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

      1. *-commutative 90.3%

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

      2. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

   8. Applied egg-rr 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+36} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi \cdot \frac{\ell}{F}}{F}\\ \end{array} \]

Alternative 3: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+33} \lor \neg \left(\ell \leq 290000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -3.6e+33) (not (<= l 290000000000.0)))
   (* PI l)
   (- (* PI l) (* (/ l F) (/ PI F)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -3.6e+33) || !(l <= 290000000000.0)) {
		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 ((l <= -3.6e+33) || !(l <= 290000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((l / F) * (Math.PI / F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -3.6e+33) or not (l <= 290000000000.0):
		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 ((l <= -3.6e+33) || !(l <= 290000000000.0))
		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 ((l <= -3.6e+33) || ~((l <= 290000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((l / F) * (pi / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -3.6e+33], N[Not[LessEqual[l, 290000000000.0]], $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 l < -3.6000000000000003e33 or 2.9e11 < l

   1. Initial program 63.5%

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

      1. sqr-neg 63.5%

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

      2. associate-*l/ 63.5%

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

      3. *-lft-identity 63.5%

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

      4. sqr-neg 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in l around 0 48.7%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -3.6000000000000003e33 < l < 2.9e11

   1. Initial program 90.3%

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

      1. sqr-neg 90.3%

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

      2. associate-*l/ 90.3%

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

      3. *-lft-identity 90.3%

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

      4. sqr-neg 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in l around 0 90.3%

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

      1. *-commutative 90.3%

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

      2. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+33} \lor \neg \left(\ell \leq 290000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \]

Alternative 4: 91.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+33} \lor \neg \left(\ell \leq 290000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 - {F}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -3.6e+33) (not (<= l 290000000000.0)))
   (* PI l)
   (* (* PI l) (- 1.0 (pow F -2.0)))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -3.6e+33) || !(l <= 290000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) * (1.0 - pow(F, -2.0));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -3.6e+33) || !(l <= 290000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) * (1.0 - Math.pow(F, -2.0));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -3.6e+33) or not (l <= 290000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) * (1.0 - math.pow(F, -2.0))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -3.6e+33) || !(l <= 290000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) * Float64(1.0 - (F ^ -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -3.6e+33) || ~((l <= 290000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) * (1.0 - (F ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -3.6e+33], N[Not[LessEqual[l, 290000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] * N[(1.0 - N[Power[F, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.6000000000000003e33 or 2.9e11 < l

   1. Initial program 63.5%

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

      1. sqr-neg 63.5%

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

      2. associate-*l/ 63.5%

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

      3. *-lft-identity 63.5%

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

      4. sqr-neg 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in l around 0 48.7%

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

   5. Taylor expanded in F around inf 99.7%

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

   if -3.6000000000000003e33 < l < 2.9e11

   1. Initial program 90.3%

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

      1. sqr-neg 90.3%

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

      2. associate-*l/ 90.3%

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

      3. *-lft-identity 90.3%

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

      4. sqr-neg 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in l around 0 90.3%

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

   5. Taylor expanded in l around 0 90.3%

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

      1. *-lft-identity 90.3%

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

      2. associate-*l/ 90.3%

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

      3. /-rgt-identity 90.3%

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

      4. unpow2 90.3%

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

      5. associate-/l* 90.3%

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

      6. associate-/r/ 90.2%

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

      7. unpow-1 90.2%

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

      8. unpow-1 90.2%

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

      9. pow-sqr 90.3%

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

      10. metadata-eval 90.3%

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

      11. *-commutative 90.3%

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

      12. distribute-lft-out-- 90.3%

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

      13. associate-*r* 90.3%

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

      14. *-commutative 90.3%

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

      15. *-commutative 90.3%

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

      16. *-rgt-identity 90.3%

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

      17. distribute-lft-out-- 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+33} \lor \neg \left(\ell \leq 290000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \ell\right) \cdot \left(1 - {F}^{-2}\right)\\ \end{array} \]

Alternative 5: 72.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-36} \lor \neg \left(\ell \leq -1.8 \cdot 10^{-157}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(-\pi\right)}{{F}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -4.5e-36) (not (<= l -1.8e-157)))
   (* PI l)
   (/ (* l (- PI)) (pow F 2.0))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -4.5e-36) || !(l <= -1.8e-157)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (l * -((double) M_PI)) / pow(F, 2.0);
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -4.5e-36) || !(l <= -1.8e-157)) {
		tmp = Math.PI * l;
	} else {
		tmp = (l * -Math.PI) / Math.pow(F, 2.0);
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -4.5e-36) or not (l <= -1.8e-157):
		tmp = math.pi * l
	else:
		tmp = (l * -math.pi) / math.pow(F, 2.0)
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -4.5e-36) || !(l <= -1.8e-157))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(l * Float64(-pi)) / (F ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -4.5e-36) || ~((l <= -1.8e-157)))
		tmp = pi * l;
	else
		tmp = (l * -pi) / (F ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -4.5e-36], N[Not[LessEqual[l, -1.8e-157]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(l * (-Pi)), $MachinePrecision] / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.50000000000000024e-36 or -1.8e-157 < l

   1. Initial program 76.6%

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

      1. sqr-neg 76.6%

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

      2. associate-*l/ 76.6%

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

      3. *-lft-identity 76.6%

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

      4. sqr-neg 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in l around 0 69.0%

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

   5. Taylor expanded in F around inf 80.4%

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

   if -4.50000000000000024e-36 < l < -1.8e-157

   1. Initial program 90.6%

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

      1. sqr-neg 90.6%

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

      2. associate-*l/ 90.6%

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

      3. *-lft-identity 90.6%

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

      4. sqr-neg 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in l around 0 90.6%

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

   5. Taylor expanded in F around 0 57.5%

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

      1. mul-1-neg 57.5%

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

   7. Simplified 57.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-36} \lor \neg \left(\ell \leq -1.8 \cdot 10^{-157}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(-\pi\right)}{{F}^{2}}\\ \end{array} \]

Alternative 6: 72.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{-40} \lor \neg \left(\ell \leq -6.6 \cdot 10^{-158}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell \cdot \left(-{F}^{-2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= l -1.65e-40) (not (<= l -6.6e-158)))
   (* PI l)
   (* PI (* l (- (pow F -2.0))))))

C

double code(double F, double l) {
	double tmp;
	if ((l <= -1.65e-40) || !(l <= -6.6e-158)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = ((double) M_PI) * (l * -pow(F, -2.0));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((l <= -1.65e-40) || !(l <= -6.6e-158)) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.PI * (l * -Math.pow(F, -2.0));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (l <= -1.65e-40) or not (l <= -6.6e-158):
		tmp = math.pi * l
	else:
		tmp = math.pi * (l * -math.pow(F, -2.0))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if ((l <= -1.65e-40) || !(l <= -6.6e-158))
		tmp = Float64(pi * l);
	else
		tmp = Float64(pi * Float64(l * Float64(-(F ^ -2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((l <= -1.65e-40) || ~((l <= -6.6e-158)))
		tmp = pi * l;
	else
		tmp = pi * (l * -(F ^ -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[Or[LessEqual[l, -1.65e-40], N[Not[LessEqual[l, -6.6e-158]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(Pi * N[(l * (-N[Power[F, -2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.64999999999999996e-40 or -6.6000000000000003e-158 < l

   1. Initial program 76.6%

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

      1. sqr-neg 76.6%

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

      2. associate-*l/ 76.6%

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

      3. *-lft-identity 76.6%

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

      4. sqr-neg 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in l around 0 69.0%

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

   5. Taylor expanded in F around inf 80.4%

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

   if -1.64999999999999996e-40 < l < -6.6000000000000003e-158

   1. Initial program 90.6%

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

      1. sqr-neg 90.6%

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

      2. associate-*l/ 90.6%

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

      3. *-lft-identity 90.6%

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

      4. sqr-neg 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in l around 0 90.6%

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

      1. *-commutative 90.6%

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

      2. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in F around 0 57.5%

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

       1. associate-*r/ 57.5%

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

       2. *-commutative 57.5%

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

       3. unpow2 57.5%

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

       4. times-frac 66.6%

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

       5. metadata-eval 66.6%

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

       6. distribute-neg-frac 66.6%

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

       7. distribute-lft-neg-out 66.6%

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

       8. associate-*r/ 66.5%

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

       9. *-lft-identity 66.5%

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

       10. associate-*l/ 66.4%

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

       11. associate-*r* 57.4%

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

       12. unpow-1 57.4%

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

       13. unpow-1 57.4%

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

       14. pow-sqr 57.4%

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

       15. metadata-eval 57.4%

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

       16. *-commutative 57.4%

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

       17. associate-*r* 57.5%

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

       18. distribute-rgt-neg-in 57.5%

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

       19. distribute-rgt-neg-in 57.5%

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

   11. Simplified 57.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{-40} \lor \neg \left(\ell \leq -6.6 \cdot 10^{-158}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell \cdot \left(-{F}^{-2}\right)\right)\\ \end{array} \]

Alternative 7: 73.5% 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 78.2%

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

   1. sqr-neg 78.2%

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

   2. associate-*l/ 78.2%

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

   3. *-lft-identity 78.2%

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

   4. sqr-neg 78.2%

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

3. Simplified 78.2%

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

4. Taylor expanded in l around 0 71.4%

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

5. Taylor expanded in F around inf 75.3%

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

6. Final simplification 75.3%

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

Reproduce

herbie shell --seed 2023333 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))