VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 98.2%

Time: 26.7s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. F = -2.0199706030786085e-105
  2. l = -412659668963807.5

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% 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: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\right)}^{2}, 1\right)}}{F} \cdot \frac{1}{F} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (-
  (* PI l)
  (*
   (/ (/ (sin (* PI l)) (fma -0.5 (pow (log1p (expm1 (* PI l))) 2.0) 1.0)) F)
   (/ 1.0 F))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - (((sin((((double) M_PI) * l)) / fma(-0.5, pow(log1p(expm1((((double) M_PI) * l))), 2.0), 1.0)) / F) * (1.0 / F));
}

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(Float64(sin(Float64(pi * l)) / fma(-0.5, (log1p(expm1(Float64(pi * l))) ^ 2.0), 1.0)) / F) * Float64(1.0 / F)))
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(N[Sin[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(-0.5 * N[Power[N[Log[1 + N[(Exp[N[(Pi * l), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. sqr-neg 79.5%

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

   2. associate-*l/ 79.6%

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

   3. *-lft-identity 79.6%

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

   4. sqr-neg 79.6%

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

3. Simplified 79.6%

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

   1. tan-quot 79.6%

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

   2. clear-num 79.6%

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

5. Applied egg-rr 79.6%

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

6. Taylor expanded in l around 0 83.2%

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

   1. +-commutative 83.2%

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

   2. fma-def 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\ell}^{2} \cdot {\pi}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   3. *-commutative 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\pi}^{2} \cdot {\ell}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   4. unpow2 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \pi\right)} \cdot {\ell}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   5. unpow2 83.2%

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

   6. swap-sqr 83.2%

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

   7. unpow2 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   8. *-commutative 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\ell \cdot \pi\right)}}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

8. Simplified 83.2%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]
9. Step-by-step derivation

   1. associate-/r* 93.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F}}{F}} \]

   2. div-inv 93.4%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F} \cdot \frac{1}{F}} \]

   3. clear-num 93.3%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}}{F} \cdot \frac{1}{F} \]

   4. *-commutative 93.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \color{blue}{\left(\ell \cdot \pi\right)}}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{1}{F} \]

10. Applied egg-rr 93.3%

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{1}{F}} \]
11. Step-by-step derivation

    1. *-commutative 93.3%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\pi \cdot \ell\right)}}^{2}, 1\right)}}{F} \cdot \frac{1}{F} \]

    2. log1p-expm1-u 98.6%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\right)}}^{2}, 1\right)}}{F} \cdot \frac{1}{F} \]

12. Applied egg-rr 98.6%

    \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\right)}}^{2}, 1\right)}}{F} \cdot \frac{1}{F} \]

13. Final simplification 98.6%

    \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\right)}^{2}, 1\right)}}{F} \cdot \frac{1}{F} \]

Alternative 2: 92.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+48} \lor \neg \left(\pi \cdot \ell \leq 5000000\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{-2}{F \cdot {\left(\pi \cdot \ell\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{-F}}{F}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+48) (not (<= (* PI l) 5000000.0)))
   (- (* PI l) (* (/ (sin (* PI l)) F) (/ -2.0 (* F (pow (* PI l) 2.0)))))
   (fma PI l (/ (/ (tan (* PI l)) (- F)) F))))

C

double code(double F, double l) {
	double tmp;
	if (((((double) M_PI) * l) <= -1e+48) || !((((double) M_PI) * l) <= 5000000.0)) {
		tmp = (((double) M_PI) * l) - ((sin((((double) M_PI) * l)) / F) * (-2.0 / (F * pow((((double) M_PI) * l), 2.0))));
	} else {
		tmp = fma(((double) M_PI), l, ((tan((((double) M_PI) * l)) / -F) / F));
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if ((Float64(pi * l) <= -1e+48) || !(Float64(pi * l) <= 5000000.0))
		tmp = Float64(Float64(pi * l) - Float64(Float64(sin(Float64(pi * l)) / F) * Float64(-2.0 / Float64(F * (Float64(pi * l) ^ 2.0)))));
	else
		tmp = fma(pi, l, Float64(Float64(tan(Float64(pi * l)) / Float64(-F)) / F));
	end
	return tmp
end

Wolfram

code[F_, l_] := If[Or[LessEqual[N[(Pi * l), $MachinePrecision], -1e+48], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 5000000.0]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Sin[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] * N[(-2.0 / N[(F * N[Power[N[(Pi * l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Pi * l + N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / (-F)), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1.00000000000000004e48 or 5e6 < (*.f64 (PI.f64) l)

   1. Initial program 63.4%

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

      1. sqr-neg 63.4%

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

      2. associate-*l/ 63.4%

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

      3. *-lft-identity 63.4%

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

      4. sqr-neg 63.4%

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

   3. Simplified 63.4%

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

      1. tan-quot 63.4%

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

      2. clear-num 63.4%

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

   5. Applied egg-rr 63.4%

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

   6. Taylor expanded in l around 0 72.9%

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

      1. +-commutative 72.9%

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

      2. fma-def 72.9%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\ell}^{2} \cdot {\pi}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      3. *-commutative 72.9%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\pi}^{2} \cdot {\ell}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      4. unpow2 72.9%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \pi\right)} \cdot {\ell}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      5. unpow2 72.9%

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

      6. swap-sqr 72.9%

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

      7. unpow2 72.9%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      8. *-commutative 72.9%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\ell \cdot \pi\right)}}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   8. Simplified 72.9%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]
   9. Step-by-step derivation

      1. associate-/r/ 72.9%

         \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)} \cdot \sin \left(\pi \cdot \ell\right)}}{F \cdot F} \]

      2. times-frac 90.7%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}} \]

      3. *-commutative 90.7%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{\sin \color{blue}{\left(\ell \cdot \pi\right)}}{F} \]

   10. Applied egg-rr 90.7%

       \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{\sin \left(\ell \cdot \pi\right)}{F}} \]

   11. Taylor expanded in l around inf 90.7%

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

       1. unpow2 90.7%

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

       2. unpow2 90.7%

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

       3. swap-sqr 90.7%

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

       4. unpow2 90.7%

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

   13. Simplified 90.7%

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

   if -1.00000000000000004e48 < (*.f64 (PI.f64) l) < 5e6

   1. Initial program 93.7%

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

      1. *-commutative 93.7%

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

      2. sqr-neg 93.7%

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

      3. *-commutative 93.7%

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

      4. fma-neg 93.7%

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

      5. associate-*l/ 93.8%

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

      6. times-frac 97.1%

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

      7. distribute-lft-neg-in 97.1%

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

      8. neg-mul-1 97.1%

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

      9. associate-/r* 97.1%

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

      10. metadata-eval 97.1%

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

      11. distribute-neg-frac 97.1%

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

      12. metadata-eval 97.1%

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

      13. times-frac 93.8%

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

   3. Simplified 97.1%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+48} \lor \neg \left(\pi \cdot \ell \leq 5000000\right):\\ \;\;\;\;\pi \cdot \ell - \frac{\sin \left(\pi \cdot \ell\right)}{F} \cdot \frac{-2}{F \cdot {\left(\pi \cdot \ell\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi, \ell, \frac{\frac{\tan \left(\pi \cdot \ell\right)}{-F}}{F}\right)\\ \end{array} \]

Alternative 3: 94.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell + \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, \sqrt{{\left(\pi \cdot \ell\right)}^{4}}, 1\right)}}{F} \cdot \frac{-1}{F} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (+
  (* PI l)
  (*
   (/ (/ (sin (* PI l)) (fma -0.5 (sqrt (pow (* PI l) 4.0)) 1.0)) F)
   (/ -1.0 F))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) + (((sin((((double) M_PI) * l)) / fma(-0.5, sqrt(pow((((double) M_PI) * l), 4.0)), 1.0)) / F) * (-1.0 / F));
}

Julia

function code(F, l)
	return Float64(Float64(pi * l) + Float64(Float64(Float64(sin(Float64(pi * l)) / fma(-0.5, sqrt((Float64(pi * l) ^ 4.0)), 1.0)) / F) * Float64(-1.0 / F)))
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[(N[Sin[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(-0.5 * N[Sqrt[N[Power[N[(Pi * l), $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. sqr-neg 79.5%

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

   2. associate-*l/ 79.6%

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

   3. *-lft-identity 79.6%

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

   4. sqr-neg 79.6%

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

3. Simplified 79.6%

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

   1. tan-quot 79.6%

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

   2. clear-num 79.6%

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

5. Applied egg-rr 79.6%

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

6. Taylor expanded in l around 0 83.2%

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

   1. +-commutative 83.2%

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

   2. fma-def 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\ell}^{2} \cdot {\pi}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   3. *-commutative 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\pi}^{2} \cdot {\ell}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   4. unpow2 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \pi\right)} \cdot {\ell}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   5. unpow2 83.2%

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

   6. swap-sqr 83.2%

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

   7. unpow2 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   8. *-commutative 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\ell \cdot \pi\right)}}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

8. Simplified 83.2%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]
9. Step-by-step derivation

   1. associate-/r* 93.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F}}{F}} \]

   2. div-inv 93.4%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F} \cdot \frac{1}{F}} \]

   3. clear-num 93.3%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}}{F} \cdot \frac{1}{F} \]

   4. *-commutative 93.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \color{blue}{\left(\ell \cdot \pi\right)}}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{1}{F} \]

10. Applied egg-rr 93.3%

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{1}{F}} \]
11. Step-by-step derivation

    1. add-sqr-sqrt 93.3%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, \color{blue}{\sqrt{{\left(\ell \cdot \pi\right)}^{2}} \cdot \sqrt{{\left(\ell \cdot \pi\right)}^{2}}}, 1\right)}}{F} \cdot \frac{1}{F} \]

    2. sqrt-unprod 96.0%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, \color{blue}{\sqrt{{\left(\ell \cdot \pi\right)}^{2} \cdot {\left(\ell \cdot \pi\right)}^{2}}}, 1\right)}}{F} \cdot \frac{1}{F} \]

    3. *-commutative 96.0%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, \sqrt{{\color{blue}{\left(\pi \cdot \ell\right)}}^{2} \cdot {\left(\ell \cdot \pi\right)}^{2}}, 1\right)}}{F} \cdot \frac{1}{F} \]

    4. *-commutative 96.0%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, \sqrt{{\left(\pi \cdot \ell\right)}^{2} \cdot {\color{blue}{\left(\pi \cdot \ell\right)}}^{2}}, 1\right)}}{F} \cdot \frac{1}{F} \]

    5. pow-prod-up 96.0%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, \sqrt{\color{blue}{{\left(\pi \cdot \ell\right)}^{\left(2 + 2\right)}}}, 1\right)}}{F} \cdot \frac{1}{F} \]

    6. *-commutative 96.0%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, \sqrt{{\color{blue}{\left(\ell \cdot \pi\right)}}^{\left(2 + 2\right)}}, 1\right)}}{F} \cdot \frac{1}{F} \]

    7. metadata-eval 96.0%

       \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, \sqrt{{\left(\ell \cdot \pi\right)}^{\color{blue}{4}}}, 1\right)}}{F} \cdot \frac{1}{F} \]

12. Applied egg-rr 96.0%

    \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\ell \cdot \pi\right)}{\mathsf{fma}\left(-0.5, \color{blue}{\sqrt{{\left(\ell \cdot \pi\right)}^{4}}}, 1\right)}}{F} \cdot \frac{1}{F} \]

13. Final simplification 96.0%

    \[\leadsto \pi \cdot \ell + \frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, \sqrt{{\left(\pi \cdot \ell\right)}^{4}}, 1\right)}}{F} \cdot \frac{-1}{F} \]

Alternative 4: 91.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (-
  (* PI l)
  (* (/ (/ 1.0 (fma -0.5 (pow (* PI l) 2.0) 1.0)) F) (/ (sin (* PI l)) F))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - (((1.0 / fma(-0.5, pow((((double) M_PI) * l), 2.0), 1.0)) / F) * (sin((((double) M_PI) * l)) / F));
}

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(Float64(1.0 / fma(-0.5, (Float64(pi * l) ^ 2.0), 1.0)) / F) * Float64(sin(Float64(pi * l)) / F)))
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(1.0 / N[(-0.5 * N[Power[N[(Pi * l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] * N[(N[Sin[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. sqr-neg 79.5%

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

   2. associate-*l/ 79.6%

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

   3. *-lft-identity 79.6%

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

   4. sqr-neg 79.6%

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

3. Simplified 79.6%

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

   1. tan-quot 79.6%

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

   2. clear-num 79.6%

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

5. Applied egg-rr 79.6%

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

6. Taylor expanded in l around 0 83.2%

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

   1. +-commutative 83.2%

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

   2. fma-def 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\ell}^{2} \cdot {\pi}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   3. *-commutative 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\pi}^{2} \cdot {\ell}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   4. unpow2 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \pi\right)} \cdot {\ell}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   5. unpow2 83.2%

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

   6. swap-sqr 83.2%

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

   7. unpow2 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   8. *-commutative 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\ell \cdot \pi\right)}}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

8. Simplified 83.2%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]
9. Step-by-step derivation

   1. associate-/r/ 83.2%

      \[\leadsto \pi \cdot \ell - \frac{\color{blue}{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)} \cdot \sin \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   2. times-frac 93.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F}} \]

   3. *-commutative 93.3%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{\sin \color{blue}{\left(\ell \cdot \pi\right)}}{F} \]

10. Applied egg-rr 93.3%

    \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{F} \cdot \frac{\sin \left(\ell \cdot \pi\right)}{F}} \]

11. Final simplification 93.3%

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)}}{F} \cdot \frac{\sin \left(\pi \cdot \ell\right)}{F} \]

Alternative 5: 40.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\sqrt{F}}}{\sqrt{F}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{\ell \cdot \left(\pi \cdot -0.5 - \pi \cdot -0.16666666666666666\right) + \frac{1}{\pi \cdot \ell}}}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 1.1e-219)
   (- (* PI l) (/ (/ (/ (tan (* PI l)) F) (sqrt F)) (sqrt F)))
   (+
    (* PI l)
    (/
     (/
      -1.0
      (+ (* l (- (* PI -0.5) (* PI -0.16666666666666666))) (/ 1.0 (* PI l))))
     (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 1.1e-219) {
		tmp = (((double) M_PI) * l) - (((tan((((double) M_PI) * l)) / F) / sqrt(F)) / sqrt(F));
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / ((l * ((((double) M_PI) * -0.5) - (((double) M_PI) * -0.16666666666666666))) + (1.0 / (((double) M_PI) * l)))) / (F * F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (F <= 1.1e-219) {
		tmp = (Math.PI * l) - (((Math.tan((Math.PI * l)) / F) / Math.sqrt(F)) / Math.sqrt(F));
	} else {
		tmp = (Math.PI * l) + ((-1.0 / ((l * ((Math.PI * -0.5) - (Math.PI * -0.16666666666666666))) + (1.0 / (Math.PI * l)))) / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 1.1e-219:
		tmp = (math.pi * l) - (((math.tan((math.pi * l)) / F) / math.sqrt(F)) / math.sqrt(F))
	else:
		tmp = (math.pi * l) + ((-1.0 / ((l * ((math.pi * -0.5) - (math.pi * -0.16666666666666666))) + (1.0 / (math.pi * l)))) / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 1.1e-219)
		tmp = Float64(Float64(pi * l) - Float64(Float64(Float64(tan(Float64(pi * l)) / F) / sqrt(F)) / sqrt(F)));
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / Float64(Float64(l * Float64(Float64(pi * -0.5) - Float64(pi * -0.16666666666666666))) + Float64(1.0 / Float64(pi * l)))) / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 1.1e-219)
		tmp = (pi * l) - (((tan((pi * l)) / F) / sqrt(F)) / sqrt(F));
	else
		tmp = (pi * l) + ((-1.0 / ((l * ((pi * -0.5) - (pi * -0.16666666666666666))) + (1.0 / (pi * l)))) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 1.1e-219], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / N[Sqrt[F], $MachinePrecision]), $MachinePrecision] / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / N[(N[(l * N[(N[(Pi * -0.5), $MachinePrecision] - N[(Pi * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(Pi * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.1e-219

   1. Initial program 74.1%

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

      1. associate-*l/ 74.1%

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

      2. *-un-lft-identity 74.1%

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

      3. associate-/r* 77.0%

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

      4. add-sqr-sqrt 2.7%

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

      5. associate-/r* 2.7%

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

   3. Applied egg-rr 2.7%

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

   if 1.1e-219 < F

   1. Initial program 88.2%

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

      1. sqr-neg 88.2%

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

      2. associate-*l/ 88.3%

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

      3. *-lft-identity 88.3%

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

      4. sqr-neg 88.3%

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

   3. Simplified 88.3%

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

      1. tan-quot 88.3%

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

      2. clear-num 88.3%

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

   5. Applied egg-rr 88.3%

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

   6. Taylor expanded in l around 0 91.0%

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

      1. +-commutative 91.0%

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

      2. fma-def 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\ell}^{2} \cdot {\pi}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      3. *-commutative 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\pi}^{2} \cdot {\ell}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      4. unpow2 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \pi\right)} \cdot {\ell}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      5. unpow2 91.0%

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

      6. swap-sqr 91.0%

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

      7. unpow2 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      8. *-commutative 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\ell \cdot \pi\right)}}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   8. Simplified 91.0%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   9. Taylor expanded in l around 0 90.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\color{blue}{\ell \cdot \left(-0.5 \cdot \pi - -0.16666666666666666 \cdot \pi\right) + \frac{1}{\ell \cdot \pi}}}}{F \cdot F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{\sqrt{F}}}{\sqrt{F}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{\ell \cdot \left(\pi \cdot -0.5 - \pi \cdot -0.16666666666666666\right) + \frac{1}{\pi \cdot \ell}}}{F \cdot F}\\ \end{array} \]

Alternative 6: 81.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{\ell \cdot \left(\pi \cdot -0.5 - \pi \cdot -0.16666666666666666\right) + \frac{1}{\pi \cdot \ell}}}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 1.1e-219)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))
   (+
    (* PI l)
    (/
     (/
      -1.0
      (+ (* l (- (* PI -0.5) (* PI -0.16666666666666666))) (/ 1.0 (* PI l))))
     (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 1.1e-219) {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / ((l * ((((double) M_PI) * -0.5) - (((double) M_PI) * -0.16666666666666666))) + (1.0 / (((double) M_PI) * l)))) / (F * F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if (F <= 1.1e-219) {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	} else {
		tmp = (Math.PI * l) + ((-1.0 / ((l * ((Math.PI * -0.5) - (Math.PI * -0.16666666666666666))) + (1.0 / (Math.PI * l)))) / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 1.1e-219:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	else:
		tmp = (math.pi * l) + ((-1.0 / ((l * ((math.pi * -0.5) - (math.pi * -0.16666666666666666))) + (1.0 / (math.pi * l)))) / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 1.1e-219)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / Float64(Float64(l * Float64(Float64(pi * -0.5) - Float64(pi * -0.16666666666666666))) + Float64(1.0 / Float64(pi * l)))) / Float64(F * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (F <= 1.1e-219)
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	else
		tmp = (pi * l) + ((-1.0 / ((l * ((pi * -0.5) - (pi * -0.16666666666666666))) + (1.0 / (pi * l)))) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 1.1e-219], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / N[(N[(l * N[(N[(Pi * -0.5), $MachinePrecision] - N[(Pi * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(Pi * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.1e-219

   1. Initial program 74.1%

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

      1. associate-*l/ 74.1%

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

      2. *-un-lft-identity 74.1%

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

      3. associate-/r* 77.0%

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

   3. Applied egg-rr 77.0%

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

   if 1.1e-219 < F

   1. Initial program 88.2%

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

      1. sqr-neg 88.2%

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

      2. associate-*l/ 88.3%

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

      3. *-lft-identity 88.3%

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

      4. sqr-neg 88.3%

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

   3. Simplified 88.3%

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

      1. tan-quot 88.3%

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

      2. clear-num 88.3%

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

   5. Applied egg-rr 88.3%

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

   6. Taylor expanded in l around 0 91.0%

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

      1. +-commutative 91.0%

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

      2. fma-def 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\ell}^{2} \cdot {\pi}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      3. *-commutative 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\pi}^{2} \cdot {\ell}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      4. unpow2 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \pi\right)} \cdot {\ell}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      5. unpow2 91.0%

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

      6. swap-sqr 91.0%

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

      7. unpow2 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      8. *-commutative 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\ell \cdot \pi\right)}}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   8. Simplified 91.0%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   9. Taylor expanded in l around 0 90.2%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\color{blue}{\ell \cdot \left(-0.5 \cdot \pi - -0.16666666666666666 \cdot \pi\right) + \frac{1}{\ell \cdot \pi}}}}{F \cdot F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{\ell \cdot \left(\pi \cdot -0.5 - \pi \cdot -0.16666666666666666\right) + \frac{1}{\pi \cdot \ell}}}{F \cdot F}\\ \end{array} \]

Alternative 7: 81.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \frac{\frac{1}{\ell}}{\pi}\right)}}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 1.1e-219)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))
   (+
    (* PI l)
    (/ (/ -1.0 (fma l (* PI -0.3333333333333333) (/ (/ 1.0 l) PI))) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 1.1e-219) {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / fma(l, (((double) M_PI) * -0.3333333333333333), ((1.0 / l) / ((double) M_PI)))) / (F * F));
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if (F <= 1.1e-219)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / fma(l, Float64(pi * -0.3333333333333333), Float64(Float64(1.0 / l) / pi))) / Float64(F * F)));
	end
	return tmp
end

Wolfram

code[F_, l_] := If[LessEqual[F, 1.1e-219], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / N[(l * N[(Pi * -0.3333333333333333), $MachinePrecision] + N[(N[(1.0 / l), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.1e-219

   1. Initial program 74.1%

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

      1. associate-*l/ 74.1%

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

      2. *-un-lft-identity 74.1%

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

      3. associate-/r* 77.0%

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

   3. Applied egg-rr 77.0%

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

   if 1.1e-219 < F

   1. Initial program 88.2%

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

      1. sqr-neg 88.2%

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

      2. associate-*l/ 88.3%

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

      3. *-lft-identity 88.3%

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

      4. sqr-neg 88.3%

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

   3. Simplified 88.3%

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

      1. tan-quot 88.3%

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

      2. clear-num 88.3%

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

   5. Applied egg-rr 88.3%

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

   6. Taylor expanded in l around 0 91.0%

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

      1. +-commutative 91.0%

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

      2. fma-def 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\ell}^{2} \cdot {\pi}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      3. *-commutative 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\pi}^{2} \cdot {\ell}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      4. unpow2 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{\left(\pi \cdot \pi\right)} \cdot {\ell}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      5. unpow2 91.0%

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

      6. swap-sqr 91.0%

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

      7. unpow2 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{{\left(\pi \cdot \ell\right)}^{2}}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

      8. *-commutative 91.0%

         \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\mathsf{fma}\left(-0.5, {\color{blue}{\left(\ell \cdot \pi\right)}}^{2}, 1\right)}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   8. Simplified 91.0%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, {\left(\ell \cdot \pi\right)}^{2}, 1\right)}}{\sin \left(\pi \cdot \ell\right)}}}{F \cdot F} \]

   9. Taylor expanded in l around 0 90.2%

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

       1. fma-def 90.2%

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

       2. distribute-rgt-out-- 90.2%

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

       3. metadata-eval 90.2%

          \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\mathsf{fma}\left(\ell, \pi \cdot \color{blue}{-0.3333333333333333}, \frac{1}{\ell \cdot \pi}\right)}}{F \cdot F} \]

       4. associate-/r* 90.1%

          \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \color{blue}{\frac{\frac{1}{\ell}}{\pi}}\right)}}{F \cdot F} \]

   11. Simplified 90.1%

       \[\leadsto \pi \cdot \ell - \frac{\frac{1}{\color{blue}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \frac{\frac{1}{\ell}}{\pi}\right)}}}{F \cdot F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{\mathsf{fma}\left(\ell, \pi \cdot -0.3333333333333333, \frac{\frac{1}{\ell}}{\pi}\right)}}{F \cdot F}\\ \end{array} \]

Alternative 8: 82.1% 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 79.5%

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

   1. *-commutative 79.5%

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

   2. sqr-neg 79.5%

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

   3. *-commutative 79.5%

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

   4. fma-neg 79.5%

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

   5. associate-*l/ 79.6%

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

   6. times-frac 81.3%

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

   7. distribute-lft-neg-in 81.3%

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

   8. neg-mul-1 81.3%

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

   9. associate-/r* 81.3%

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

   10. metadata-eval 81.3%

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

   11. distribute-neg-frac 81.3%

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

   12. metadata-eval 81.3%

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

   13. times-frac 79.6%

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

3. Simplified 81.3%

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

4. Final simplification 81.3%

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

Alternative 9: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{\frac{F}{\frac{\ell}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= F 1.1e-219)
   (- (* PI l) (/ PI (/ F (/ l F))))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if (F <= 1.1e-219) {
		tmp = (((double) M_PI) * l) - (((double) M_PI) / (F / (l / 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 <= 1.1e-219) {
		tmp = (Math.PI * l) - (Math.PI / (F / (l / F)));
	} else {
		tmp = (Math.PI * l) - (Math.tan((Math.PI * l)) / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if F <= 1.1e-219:
		tmp = (math.pi * l) - (math.pi / (F / (l / 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 <= 1.1e-219)
		tmp = Float64(Float64(pi * l) - Float64(pi / Float64(F / Float64(l / 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 <= 1.1e-219)
		tmp = (pi * l) - (pi / (F / (l / F)));
	else
		tmp = (pi * l) - (tan((pi * l)) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[F, 1.1e-219], N[(N[(Pi * l), $MachinePrecision] - N[(Pi / N[(F / N[(l / F), $MachinePrecision]), $MachinePrecision]), $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 < 1.1e-219

   1. Initial program 74.1%

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

      1. sqr-neg 74.1%

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

      2. associate-*l/ 74.1%

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

      3. *-lft-identity 74.1%

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

      4. sqr-neg 74.1%

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

   3. Simplified 74.1%

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

   4. Taylor expanded in l around 0 68.2%

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

      1. *-commutative 68.2%

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

      2. times-frac 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. associate-*l/ 71.1%

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

      2. associate-/l* 71.1%

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

   8. Applied egg-rr 71.1%

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

   if 1.1e-219 < F

   1. Initial program 88.2%

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

      1. sqr-neg 88.2%

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

      2. associate-*l/ 88.3%

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

      3. *-lft-identity 88.3%

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

      4. sqr-neg 88.3%

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

   3. Simplified 88.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.1 \cdot 10^{-219}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{\frac{F}{\frac{\ell}{F}}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]

Alternative 10: 82.1% 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 79.5%

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

   1. associate-*l/ 79.6%

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

   2. *-un-lft-identity 79.6%

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

   3. associate-/r* 81.3%

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

3. Applied egg-rr 81.3%

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

4. Final simplification 81.3%

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

Alternative 11: 74.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. *-commutative 79.5%

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

   2. sqr-neg 79.5%

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

   3. *-commutative 79.5%

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

   4. fma-neg 79.5%

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

   5. associate-*l/ 79.6%

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

   6. times-frac 81.3%

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

   7. distribute-lft-neg-in 81.3%

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

   8. neg-mul-1 81.3%

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

   9. associate-/r* 81.3%

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

   10. metadata-eval 81.3%

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

   11. distribute-neg-frac 81.3%

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

   12. metadata-eval 81.3%

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

   13. times-frac 79.6%

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

3. Simplified 81.3%

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

4. Taylor expanded in l around 0 74.2%

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

5. Final simplification 74.2%

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

Alternative 12: 74.7% accurate, 1.5× speedup

Math

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

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

Derivation

1. Initial program 79.5%

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

   1. sqr-neg 79.5%

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

   2. associate-*l/ 79.6%

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

   3. *-lft-identity 79.6%

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

   4. sqr-neg 79.6%

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

3. Simplified 79.6%

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

4. Taylor expanded in l around 0 72.4%

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

   1. *-commutative 72.4%

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

   2. times-frac 74.1%

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

6. Applied egg-rr 74.1%

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

7. Final simplification 74.1%

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

Alternative 13: 74.7% accurate, 1.5× speedup

Math

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

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(pi / Float64(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[(Pi / N[(F / N[(l / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

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

   1. sqr-neg 79.5%

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

   2. associate-*l/ 79.6%

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

   3. *-lft-identity 79.6%

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

   4. sqr-neg 79.6%

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

3. Simplified 79.6%

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

4. Taylor expanded in l around 0 72.4%

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

   1. *-commutative 72.4%

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

   2. times-frac 74.1%

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

6. Applied egg-rr 74.1%

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

   1. associate-*l/ 74.2%

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

   2. associate-/l* 74.2%

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

8. Applied egg-rr 74.2%

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

9. Final simplification 74.2%

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

Reproduce

herbie shell --seed 2023305 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))