VandenBroeck and Keller, Equation (6)

Percentage Accurate: 77.1% → 94.5%

Time: 31.0s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. F = 1.2498766750596967e+82
  2. l = 3892636595763351.5

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: 77.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell + \frac{\frac{-1}{F}}{\left(\frac{F}{\pi \cdot \ell} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) + {\ell}^{3} \cdot \left(F \cdot \left({\pi}^{3} \cdot -0.013888888888888888 - {\pi}^{3} \cdot 0.008333333333333333\right)\right)} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (+
  (* PI l)
  (/
   (/ -1.0 F)
   (+
    (-
     (/ F (* PI l))
     (* F (/ (* (* l (pow PI 3.0)) 0.3333333333333333) (pow PI 2.0))))
    (*
     (pow l 3.0)
     (*
      F
      (-
       (* (pow PI 3.0) -0.013888888888888888)
       (* (pow PI 3.0) 0.008333333333333333))))))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) + ((-1.0 / F) / (((F / (((double) M_PI) * l)) - (F * (((l * pow(((double) M_PI), 3.0)) * 0.3333333333333333) / pow(((double) M_PI), 2.0)))) + (pow(l, 3.0) * (F * ((pow(((double) M_PI), 3.0) * -0.013888888888888888) - (pow(((double) M_PI), 3.0) * 0.008333333333333333))))));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) + ((-1.0 / F) / (((F / (Math.PI * l)) - (F * (((l * Math.pow(Math.PI, 3.0)) * 0.3333333333333333) / Math.pow(Math.PI, 2.0)))) + (Math.pow(l, 3.0) * (F * ((Math.pow(Math.PI, 3.0) * -0.013888888888888888) - (Math.pow(Math.PI, 3.0) * 0.008333333333333333))))));
}

Python

def code(F, l):
	return (math.pi * l) + ((-1.0 / F) / (((F / (math.pi * l)) - (F * (((l * math.pow(math.pi, 3.0)) * 0.3333333333333333) / math.pow(math.pi, 2.0)))) + (math.pow(l, 3.0) * (F * ((math.pow(math.pi, 3.0) * -0.013888888888888888) - (math.pow(math.pi, 3.0) * 0.008333333333333333))))))

Julia

function code(F, l)
	return Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(Float64(Float64(F / Float64(pi * l)) - Float64(F * Float64(Float64(Float64(l * (pi ^ 3.0)) * 0.3333333333333333) / (pi ^ 2.0)))) + Float64((l ^ 3.0) * Float64(F * Float64(Float64((pi ^ 3.0) * -0.013888888888888888) - Float64((pi ^ 3.0) * 0.008333333333333333)))))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) + ((-1.0 / F) / (((F / (pi * l)) - (F * (((l * (pi ^ 3.0)) * 0.3333333333333333) / (pi ^ 2.0)))) + ((l ^ 3.0) * (F * (((pi ^ 3.0) * -0.013888888888888888) - ((pi ^ 3.0) * 0.008333333333333333))))));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(N[(N[(F / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(F * N[(N[(N[(l * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 3.0], $MachinePrecision] * N[(F * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.013888888888888888), $MachinePrecision] - N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

   1. associate-/r/ 78.8%

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

   2. associate-/l* 82.8%

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

   3. clear-num 82.8%

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

   4. add-sqr-sqrt 38.3%

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

   5. sqrt-prod 68.3%

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

   6. sqr-neg 68.3%

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

   7. sqrt-unprod 30.4%

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

   8. div-inv 30.4%

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

   9. metadata-eval 30.4%

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

   10. add-sqr-sqrt 0.0%

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

   11. sqrt-prod 40.9%

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

   12. sqrt-div 40.9%

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

   13. add-sqr-sqrt 67.4%

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

   14. associate-*l/ 67.4%

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

   15. clear-num 67.4%

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

   16. associate-*l/ 67.5%

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

   17. *-un-lft-identity 67.5%

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

3. Applied egg-rr 82.8%

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

4. Taylor expanded in l around 0 94.7%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{-1 \cdot \left({\ell}^{3} \cdot \left(-1 \cdot \frac{F \cdot {\left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}^{2}}{{\pi}^{3}} + \frac{F \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \left(-0.5 \cdot \left({\pi}^{2} \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right) + 0.041666666666666664 \cdot {\pi}^{5}\right)\right)}{{\pi}^{2}}\right)\right) + \left(-1 \cdot \frac{F \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}} + \frac{F}{\ell \cdot \pi}\right)}} \]

6. Simplified 94.7%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\left(\frac{F}{\ell \cdot \pi} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) - {\ell}^{3} \cdot \left(\frac{F}{{\pi}^{2}} \cdot \left(0.008333333333333333 \cdot {\pi}^{5} - \mathsf{fma}\left(-0.5, {\pi}^{3} \cdot \left(0.3333333333333333 \cdot {\pi}^{2}\right), {\pi}^{5} \cdot 0.041666666666666664\right)\right) - \frac{F}{{\pi}^{3}} \cdot \left({\pi}^{6} \cdot 0.1111111111111111\right)\right)}} \]

7. Taylor expanded in F around 0 94.7%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\left(\frac{F}{\ell \cdot \pi} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) - {\ell}^{3} \cdot \color{blue}{\left(F \cdot \left(0.008333333333333333 \cdot {\pi}^{3} - \left(-0.16666666666666666 \cdot {\pi}^{3} + \left(0.041666666666666664 \cdot {\pi}^{3} + 0.1111111111111111 \cdot {\pi}^{3}\right)\right)\right)\right)}} \]
8. Step-by-step derivation

   1. *-commutative 94.7%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\left(\frac{F}{\ell \cdot \pi} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) - {\ell}^{3} \cdot \left(F \cdot \left(0.008333333333333333 \cdot {\pi}^{3} - \left(\color{blue}{{\pi}^{3} \cdot -0.16666666666666666} + \left(0.041666666666666664 \cdot {\pi}^{3} + 0.1111111111111111 \cdot {\pi}^{3}\right)\right)\right)\right)} \]

   2. distribute-rgt-out 94.7%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\left(\frac{F}{\ell \cdot \pi} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) - {\ell}^{3} \cdot \left(F \cdot \left(0.008333333333333333 \cdot {\pi}^{3} - \left({\pi}^{3} \cdot -0.16666666666666666 + \color{blue}{{\pi}^{3} \cdot \left(0.041666666666666664 + 0.1111111111111111\right)}\right)\right)\right)} \]

   3. distribute-lft-out 94.7%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\left(\frac{F}{\ell \cdot \pi} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) - {\ell}^{3} \cdot \left(F \cdot \left(0.008333333333333333 \cdot {\pi}^{3} - \color{blue}{{\pi}^{3} \cdot \left(-0.16666666666666666 + \left(0.041666666666666664 + 0.1111111111111111\right)\right)}\right)\right)} \]

   4. metadata-eval 94.7%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\left(\frac{F}{\ell \cdot \pi} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) - {\ell}^{3} \cdot \left(F \cdot \left(0.008333333333333333 \cdot {\pi}^{3} - {\pi}^{3} \cdot \left(-0.16666666666666666 + \color{blue}{0.1527777777777778}\right)\right)\right)} \]

   5. metadata-eval 94.7%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\left(\frac{F}{\ell \cdot \pi} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) - {\ell}^{3} \cdot \left(F \cdot \left(0.008333333333333333 \cdot {\pi}^{3} - {\pi}^{3} \cdot \color{blue}{-0.013888888888888888}\right)\right)} \]

9. Simplified 94.7%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\left(\frac{F}{\ell \cdot \pi} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) - {\ell}^{3} \cdot \color{blue}{\left(F \cdot \left(0.008333333333333333 \cdot {\pi}^{3} - {\pi}^{3} \cdot -0.013888888888888888\right)\right)}} \]

10. Final simplification 94.7%

    \[\leadsto \pi \cdot \ell + \frac{\frac{-1}{F}}{\left(\frac{F}{\pi \cdot \ell} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}\right) + {\ell}^{3} \cdot \left(F \cdot \left({\pi}^{3} \cdot -0.013888888888888888 - {\pi}^{3} \cdot 0.008333333333333333\right)\right)} \]

Alternative 2: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+162} \lor \neg \left(\pi \cdot \ell \leq -5 \cdot 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \frac{-0.5}{\frac{{F}^{4}}{{\left(\pi \cdot \ell\right)}^{2}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+162) (not (<= (* PI l) -5e+14)))
   (+
    (* PI l)
    (/
     (/ -1.0 F)
     (-
      (/ F (* PI l))
      (* F (/ (* (* l (pow PI 3.0)) 0.3333333333333333) (pow PI 2.0))))))
   (-
    (* PI l)
    (expm1
     (fma l (/ PI (pow F 2.0)) (/ -0.5 (/ (pow F 4.0) (pow (* PI l) 2.0))))))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+162) || !((((double) M_PI) * l) <= -5e+14)) {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / ((F / (((double) M_PI) * l)) - (F * (((l * pow(((double) M_PI), 3.0)) * 0.3333333333333333) / pow(((double) M_PI), 2.0)))));
	} else {
		tmp = (((double) M_PI) * l) - expm1(fma(l, (((double) M_PI) / pow(F, 2.0)), (-0.5 / (pow(F, 4.0) / pow((((double) M_PI) * l), 2.0)))));
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -1e+162) || !(Float64(pi * l) <= -5e+14))
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(Float64(F / Float64(pi * l)) - Float64(F * Float64(Float64(Float64(l * (pi ^ 3.0)) * 0.3333333333333333) / (pi ^ 2.0))))));
	else
		tmp = Float64(Float64(pi * l) - expm1(fma(l, Float64(pi / (F ^ 2.0)), Float64(-0.5 / Float64((F ^ 4.0) / (Float64(pi * l) ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e+162], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], -5e+14]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(N[(F / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(F * N[(N[(N[(l * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(Exp[N[(l * N[(Pi / N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(N[Power[F, 4.0], $MachinePrecision] / N[Power[N[(Pi * l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -9.9999999999999994e161 or -5e14 < (*.f64 (PI.f64) l)

   1. Initial program 81.4%

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

      1. associate-/r/ 81.8%

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

      2. associate-/l* 86.5%

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

      3. clear-num 86.5%

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

      4. add-sqr-sqrt 38.5%

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

      5. sqrt-prod 70.2%

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

      6. sqr-neg 70.2%

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

      7. sqrt-unprod 32.1%

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

      8. div-inv 32.1%

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

      9. metadata-eval 32.1%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-prod 43.8%

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

      12. sqrt-div 43.8%

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

      13. add-sqr-sqrt 68.8%

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

      14. associate-*l/ 68.9%

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

      15. clear-num 68.8%

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

      16. associate-*l/ 68.9%

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

      17. *-un-lft-identity 68.9%

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

   3. Applied egg-rr 86.4%

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

   4. Taylor expanded in l around 0 92.1%

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

      1. +-commutative 92.1%

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

      2. mul-1-neg 92.1%

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

      3. unsub-neg 92.1%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\frac{F}{\ell \cdot \pi} - \frac{F \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}}}} \]

      4. *-lft-identity 92.1%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \frac{F \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{\color{blue}{1 \cdot {\pi}^{2}}}} \]

      5. times-frac 92.1%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \color{blue}{\frac{F}{1} \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}}} \]

      6. rem-square-sqrt 41.9%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \frac{\color{blue}{\sqrt{F} \cdot \sqrt{F}}}{1} \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}} \]

      7. associate-*l/ 41.9%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \color{blue}{\left(\frac{\sqrt{F}}{1} \cdot \sqrt{F}\right)} \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}} \]

      8. /-rgt-identity 41.9%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \left(\color{blue}{\sqrt{F}} \cdot \sqrt{F}\right) \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}} \]

      9. rem-square-sqrt 92.1%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \color{blue}{F} \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}} \]

   6. Simplified 92.1%

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

   if -9.9999999999999994e161 < (*.f64 (PI.f64) l) < -5e14

   1. Initial program 58.1%

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

      1. associate-/r/ 58.1%

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

      2. associate-/l* 58.1%

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

      3. clear-num 58.1%

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

      4. frac-2neg 58.1%

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

      5. add-sqr-sqrt 21.3%

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

      6. sqrt-unprod 58.0%

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

      7. sqr-neg 58.0%

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

      8. sqrt-prod 36.7%

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

      9. add-sqr-sqrt 55.6%

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

      10. distribute-neg-frac 55.6%

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

      11. associate-/r* 55.6%

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

      12. expm1-log1p-u 46.5%

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

      13. associate-/r* 46.5%

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

      14. distribute-neg-frac 46.5%

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

      15. add-sqr-sqrt 30.5%

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

      16. sqrt-prod 45.7%

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

      17. sqr-neg 45.7%

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

      18. sqrt-unprod 15.1%

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

      19. add-sqr-sqrt 51.9%

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

      20. frac-2neg 51.9%

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

   3. Applied egg-rr 51.9%

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

   4. Taylor expanded in l around 0 96.3%

      \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\color{blue}{-0.5 \cdot \frac{{\ell}^{2} \cdot {\pi}^{2}}{{F}^{4}} + \frac{\ell \cdot \pi}{{F}^{2}}}\right) \]
   5. Step-by-step derivation

      1. +-commutative 96.3%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\color{blue}{\frac{\ell \cdot \pi}{{F}^{2}} + -0.5 \cdot \frac{{\ell}^{2} \cdot {\pi}^{2}}{{F}^{4}}}\right) \]

      2. associate-*r/ 96.3%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\color{blue}{\ell \cdot \frac{\pi}{{F}^{2}}} + -0.5 \cdot \frac{{\ell}^{2} \cdot {\pi}^{2}}{{F}^{4}}\right) \]

      3. fma-def 96.3%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, -0.5 \cdot \frac{{\ell}^{2} \cdot {\pi}^{2}}{{F}^{4}}\right)}\right) \]

      4. associate-*r/ 96.3%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \color{blue}{\frac{-0.5 \cdot \left({\ell}^{2} \cdot {\pi}^{2}\right)}{{F}^{4}}}\right)\right) \]

      5. *-commutative 96.3%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \frac{-0.5 \cdot \color{blue}{\left({\pi}^{2} \cdot {\ell}^{2}\right)}}{{F}^{4}}\right)\right) \]

      6. unpow2 96.3%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \frac{-0.5 \cdot \left(\color{blue}{\left(\pi \cdot \pi\right)} \cdot {\ell}^{2}\right)}{{F}^{4}}\right)\right) \]

      7. unpow2 96.3%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \frac{-0.5 \cdot \left(\left(\pi \cdot \pi\right) \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)}{{F}^{4}}\right)\right) \]

      8. swap-sqr 96.3%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \frac{-0.5 \cdot \color{blue}{\left(\left(\pi \cdot \ell\right) \cdot \left(\pi \cdot \ell\right)\right)}}{{F}^{4}}\right)\right) \]

      9. unpow2 96.3%

         \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \frac{-0.5 \cdot \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}}{{F}^{4}}\right)\right) \]

      10. associate-/l* 96.3%

          \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \color{blue}{\frac{-0.5}{\frac{{F}^{4}}{{\left(\pi \cdot \ell\right)}^{2}}}}\right)\right) \]

      11. *-commutative 96.3%

          \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \frac{-0.5}{\frac{{F}^{4}}{{\color{blue}{\left(\ell \cdot \pi\right)}}^{2}}}\right)\right) \]

   6. Simplified 96.3%

      \[\leadsto \pi \cdot \ell - \mathsf{expm1}\left(\color{blue}{\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \frac{-0.5}{\frac{{F}^{4}}{{\left(\ell \cdot \pi\right)}^{2}}}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+162} \lor \neg \left(\pi \cdot \ell \leq -5 \cdot 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \mathsf{expm1}\left(\mathsf{fma}\left(\ell, \frac{\pi}{{F}^{2}}, \frac{-0.5}{\frac{{F}^{4}}{{\left(\pi \cdot \ell\right)}^{2}}}\right)\right)\\ \end{array} \]

Alternative 3: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\pi \cdot \ell} - F \cdot \frac{\left(\ell \cdot {\pi}^{3}\right) \cdot 0.3333333333333333}{{\pi}^{2}}} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (+
  (* PI l)
  (/
   (/ -1.0 F)
   (-
    (/ F (* PI l))
    (* F (/ (* (* l (pow PI 3.0)) 0.3333333333333333) (pow PI 2.0)))))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) + ((-1.0 / F) / ((F / (((double) M_PI) * l)) - (F * (((l * pow(((double) M_PI), 3.0)) * 0.3333333333333333) / pow(((double) M_PI), 2.0)))));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) + ((-1.0 / F) / ((F / (Math.PI * l)) - (F * (((l * Math.pow(Math.PI, 3.0)) * 0.3333333333333333) / Math.pow(Math.PI, 2.0)))));
}

Python

def code(F, l):
	return (math.pi * l) + ((-1.0 / F) / ((F / (math.pi * l)) - (F * (((l * math.pow(math.pi, 3.0)) * 0.3333333333333333) / math.pow(math.pi, 2.0)))))

Julia

function code(F, l)
	return Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(Float64(F / Float64(pi * l)) - Float64(F * Float64(Float64(Float64(l * (pi ^ 3.0)) * 0.3333333333333333) / (pi ^ 2.0))))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) + ((-1.0 / F) / ((F / (pi * l)) - (F * (((l * (pi ^ 3.0)) * 0.3333333333333333) / (pi ^ 2.0)))));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(N[(F / N[(Pi * l), $MachinePrecision]), $MachinePrecision] - N[(F * N[(N[(N[(l * N[Power[Pi, 3.0], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

   1. associate-/r/ 78.8%

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

   2. associate-/l* 82.8%

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

   3. clear-num 82.8%

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

   4. add-sqr-sqrt 38.3%

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

   5. sqrt-prod 68.3%

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

   6. sqr-neg 68.3%

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

   7. sqrt-unprod 30.4%

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

   8. div-inv 30.4%

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

   9. metadata-eval 30.4%

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

   10. add-sqr-sqrt 0.0%

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

   11. sqrt-prod 40.9%

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

   12. sqrt-div 40.9%

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

   13. add-sqr-sqrt 67.4%

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

   14. associate-*l/ 67.4%

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

   15. clear-num 67.4%

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

   16. associate-*l/ 67.5%

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

   17. *-un-lft-identity 67.5%

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

3. Applied egg-rr 82.8%

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

4. Taylor expanded in l around 0 88.9%

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

   1. +-commutative 88.9%

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

   2. mul-1-neg 88.9%

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

   3. unsub-neg 88.9%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\frac{F}{\ell \cdot \pi} - \frac{F \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{{\pi}^{2}}}} \]

   4. *-lft-identity 88.9%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \frac{F \cdot \left(\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)\right)}{\color{blue}{1 \cdot {\pi}^{2}}}} \]

   5. times-frac 88.9%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \color{blue}{\frac{F}{1} \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}}} \]

   6. rem-square-sqrt 42.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \frac{\color{blue}{\sqrt{F} \cdot \sqrt{F}}}{1} \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}} \]

   7. associate-*l/ 42.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \color{blue}{\left(\frac{\sqrt{F}}{1} \cdot \sqrt{F}\right)} \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}} \]

   8. /-rgt-identity 42.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \left(\color{blue}{\sqrt{F}} \cdot \sqrt{F}\right) \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}} \]

   9. rem-square-sqrt 88.9%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\ell \cdot \pi} - \color{blue}{F} \cdot \frac{\ell \cdot \left(-0.16666666666666666 \cdot {\pi}^{3} - -0.5 \cdot {\pi}^{3}\right)}{{\pi}^{2}}} \]

6. Simplified 88.9%

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

7. Final simplification 88.9%

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

Alternative 4: 82.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{-F}}{F}\right) \end{array} \]

FPCore

(FPCore (F l) :precision binary64 (fma PI l (/ (/ (tan (* PI l)) (- F)) F)))

C

double code(double F, double l) {
	return fma(((double) M_PI), l, ((tan((((double) M_PI) * l)) / -F) / F));
}

Julia

function code(F, l)
	return fma(pi, l, Float64(Float64(tan(Float64(pi * l)) / Float64(-F)) / F))
end

Wolfram

code[F_, l_] := N[(Pi * l + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / (-F)), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. sqr-neg 78.4%

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

   3. *-commutative 78.4%

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

   4. fma-neg 78.4%

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

   5. associate-*l/ 78.8%

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

   6. times-frac 82.8%

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

   7. distribute-lft-neg-in 82.8%

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

   8. neg-mul-1 82.8%

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

   9. associate-/r* 82.8%

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

   10. metadata-eval 82.8%

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

   11. distribute-neg-frac 82.8%

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

   12. metadata-eval 82.8%

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

   13. times-frac 78.8%

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

3. Simplified 82.8%

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

4. Final simplification 82.8%

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

Alternative 5: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 4.2 \cdot 10^{-182}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 4.2e-182)
   (- (* PI l) (/ (/ l (/ F PI)) F))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 4.2e-182) {
		tmp = (((double) M_PI) * l) - ((l / (F / ((double) M_PI))) / F);
	} 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 (F <= 4.2e-182) {
		tmp = (Math.PI * l) - ((l / (F / Math.PI)) / F);
	} else {
		tmp = (Math.PI * l) - (Math.tan((Math.PI * l)) / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 4.2e-182:
		tmp = (math.pi * l) - ((l / (F / math.pi)) / F)
	else:
		tmp = (math.pi * l) - (math.tan((math.pi * l)) / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 4.2e-182)
		tmp = Float64(Float64(pi * l) - Float64(Float64(l / Float64(F / pi)) / F));
	else
		tmp = Float64(Float64(pi * l) - Float64(tan(Float64(pi * l)) / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 4.2e-182)
		tmp = (pi * l) - ((l / (F / pi)) / F);
	else
		tmp = (pi * l) - (tan((pi * l)) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 4.2e-182], N[(N[(Pi * l), $MachinePrecision] - N[(N[(l / N[(F / Pi), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 4.2000000000000001e-182

   1. Initial program 73.5%

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

      1. associate-*l/ 73.5%

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

      2. *-un-lft-identity 73.5%

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

      3. associate-/r* 79.4%

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

   3. Applied egg-rr 79.4%

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

   4. Taylor expanded in l around 0 73.2%

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

      1. associate-/l* 73.3%

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

   6. Simplified 73.3%

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

   if 4.2000000000000001e-182 < F

   1. Initial program 86.4%

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

      1. sqr-neg 86.4%

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

      2. associate-*l/ 87.3%

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

      3. *-lft-identity 87.3%

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

      4. sqr-neg 87.3%

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

   3. Simplified 87.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.2 \cdot 10^{-182}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 6: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F} \end{array} \]

FPCore

(FPCore (F l) :precision binary64 (- (* PI l) (/ (/ (tan (* PI l)) F) F)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.8%

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

   2. *-un-lft-identity 78.8%

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

   3. associate-/r* 82.8%

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

3. Applied egg-rr 82.8%

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

4. Final simplification 82.8%

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

Alternative 7: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi, \ell, \frac{\frac{-\ell}{\frac{F}{\pi}}}{F}\right) \end{array} \]

FPCore

(FPCore (F l) :precision binary64 (fma PI l (/ (/ (- l) (/ F PI)) F)))

C

double code(double F, double l) {
	return fma(((double) M_PI), l, ((-l / (F / ((double) M_PI))) / F));
}

Julia

function code(F, l)
	return fma(pi, l, Float64(Float64(Float64(-l) / Float64(F / pi)) / F))
end

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. *-commutative 78.4%

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

   2. sqr-neg 78.4%

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

   3. *-commutative 78.4%

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

   4. fma-neg 78.4%

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

   5. associate-*l/ 78.8%

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

   6. times-frac 82.8%

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

   7. distribute-lft-neg-in 82.8%

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

   8. neg-mul-1 82.8%

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

   9. associate-/r* 82.8%

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

   10. metadata-eval 82.8%

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

   11. distribute-neg-frac 82.8%

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

   12. metadata-eval 82.8%

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

   13. times-frac 78.8%

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

3. Simplified 82.8%

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

4. Taylor expanded in l around 0 75.2%

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

   1. mul-1-neg 75.2%

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

   2. associate-/l* 75.2%

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

6. Simplified 75.2%

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

7. Final simplification 75.2%

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

Alternative 8: 75.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. sqr-neg 78.4%

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

   2. associate-*l/ 78.8%

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

   3. *-lft-identity 78.8%

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

   4. sqr-neg 78.8%

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

3. Simplified 78.8%

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

4. Taylor expanded in l around 0 71.1%

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

   1. *-commutative 71.1%

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

   2. times-frac 75.2%

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

6. Applied egg-rr 75.2%

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

7. Final simplification 75.2%

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

Alternative 9: 75.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. sqr-neg 78.4%

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

   2. associate-*l/ 78.8%

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

   3. *-lft-identity 78.8%

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

   4. sqr-neg 78.8%

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

3. Simplified 78.8%

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

4. Taylor expanded in l around 0 71.1%

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

   1. *-commutative 71.1%

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

   2. times-frac 75.2%

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

6. Applied egg-rr 75.2%

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

   1. *-commutative 75.2%

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

   2. clear-num 75.2%

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

   3. frac-times 75.2%

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

   4. *-un-lft-identity 75.2%

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

8. Applied egg-rr 75.2%

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

9. Final simplification 75.2%

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

Alternative 10: 75.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.8%

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

   2. *-un-lft-identity 78.8%

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

   3. associate-/r* 82.8%

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

3. Applied egg-rr 82.8%

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

4. Taylor expanded in l around 0 75.1%

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

   1. associate-/l* 75.2%

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

6. Simplified 75.2%

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

7. Final simplification 75.2%

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

Reproduce

herbie shell --seed 2023311 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))