VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 98.0%

Time: 19.9s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. F = 1.2664886660491778e+202
  2. l = -2616143047174689.5

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: 98.0% accurate, 0.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 81.2%

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

   1. associate-*l/ 81.7%

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

   2. *-un-lft-identity 81.7%

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

   3. associate-/r* 85.5%

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

4. Applied egg-rr 85.5%

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

   1. tan-quot 85.5%

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

6. Applied egg-rr 85.5%

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

7. Taylor expanded in l around 0 92.5%

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

   1. +-commutative 92.5%

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

   2. fma-define 92.5%

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

   3. *-commutative 92.5%

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

   4. unpow2 92.5%

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

   5. unpow2 92.5%

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

   6. swap-sqr 92.5%

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

   7. unpow2 92.5%

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

   8. *-commutative 92.5%

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

9. Simplified 92.5%

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

    1. *-commutative 92.5%

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

    2. log1p-expm1-u 98.9%

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

11. Applied egg-rr 98.9%

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

Alternative 2: 94.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (F l)
 :precision binary64
 (-
  (* PI l)
  (/ (/ (/ (sin (* PI l)) (fma -0.5 (sqrt (pow (* PI l) 4.0)) 1.0)) F) 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) / 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) / 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] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

   1. associate-*l/ 81.7%

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

   2. *-un-lft-identity 81.7%

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

   3. associate-/r* 85.5%

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

4. Applied egg-rr 85.5%

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

   1. tan-quot 85.5%

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

6. Applied egg-rr 85.5%

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

7. Taylor expanded in l around 0 92.5%

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

   1. +-commutative 92.5%

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

   2. fma-define 92.5%

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

   3. *-commutative 92.5%

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

   4. unpow2 92.5%

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

   5. unpow2 92.5%

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

   6. swap-sqr 92.5%

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

   7. unpow2 92.5%

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

   8. *-commutative 92.5%

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

9. Simplified 92.5%

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

    1. add-sqr-sqrt 92.5%

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

    2. sqrt-unprod 94.7%

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

    3. *-commutative 94.7%

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

    4. *-commutative 94.7%

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

    5. pow-prod-up 94.7%

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

    6. metadata-eval 94.7%

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

11. Applied egg-rr 94.7%

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

Alternative 3: 78.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. associate-*l/ 81.7%

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

   2. *-un-lft-identity 81.7%

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

   3. associate-/r* 85.5%

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

4. Applied egg-rr 85.5%

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

   1. tan-quot 85.5%

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

6. Applied egg-rr 85.5%

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

7. Taylor expanded in l around 0 92.5%

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

   1. +-commutative 92.5%

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

   2. fma-define 92.5%

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

   3. *-commutative 92.5%

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

   4. unpow2 92.5%

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

   5. unpow2 92.5%

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

   6. swap-sqr 92.5%

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

   7. unpow2 92.5%

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

   8. *-commutative 92.5%

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

9. Simplified 92.5%

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

    1. add-sqr-sqrt 45.6%

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

    2. sqrt-unprod 73.4%

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

    3. pow2 73.4%

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

11. Applied egg-rr 73.4%

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

    1. unpow2 73.4%

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

    2. rem-sqrt-square 77.9%

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

    3. *-commutative 77.9%

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

13. Simplified 77.9%

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

14. Final simplification 77.9%

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

Alternative 4: 91.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(Float64(sin(Float64(pi * l)) / fma(-0.5, Float64(pi * Float64(l * Float64(pi * l))), 1.0)) / F) / 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[(Pi * N[(l * N[(Pi * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

   1. associate-*l/ 81.7%

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

   2. *-un-lft-identity 81.7%

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

   3. associate-/r* 85.5%

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

4. Applied egg-rr 85.5%

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

   1. tan-quot 85.5%

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

6. Applied egg-rr 85.5%

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

7. Taylor expanded in l around 0 92.5%

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

   1. +-commutative 92.5%

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

   2. fma-define 92.5%

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

   3. *-commutative 92.5%

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

   4. unpow2 92.5%

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

   5. unpow2 92.5%

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

   6. swap-sqr 92.5%

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

   7. unpow2 92.5%

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

   8. *-commutative 92.5%

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

9. Simplified 92.5%

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

    1. unpow2 92.5%

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

    2. *-commutative 92.5%

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

    3. associate-*r* 92.5%

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

11. Applied egg-rr 92.5%

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

12. Final simplification 92.5%

    \[\leadsto \pi \cdot \ell - \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, \pi \cdot \left(\ell \cdot \left(\pi \cdot \ell\right)\right), 1\right)}}{F}}{F} \]
13. Add Preprocessing

Alternative 5: 90.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. associate-*l/ 81.7%

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

   2. *-un-lft-identity 81.7%

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

   3. associate-/r* 85.5%

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

4. Applied egg-rr 85.5%

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

   1. tan-quot 85.5%

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

6. Applied egg-rr 85.5%

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

7. Taylor expanded in l around 0 92.5%

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

   1. +-commutative 92.5%

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

   2. fma-define 92.5%

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

   3. *-commutative 92.5%

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

   4. unpow2 92.5%

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

   5. unpow2 92.5%

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

   6. swap-sqr 92.5%

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

   7. unpow2 92.5%

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

   8. *-commutative 92.5%

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

9. Simplified 92.5%

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

10. Taylor expanded in l around 0 92.0%

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

11. Final simplification 92.0%

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

Alternative 6: 82.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. associate-*l/ 81.7%

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

   2. *-un-lft-identity 81.7%

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

   3. associate-/r* 85.5%

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

4. Applied egg-rr 85.5%

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

   1. tan-quot 85.5%

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

6. Applied egg-rr 85.5%

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

Alternative 7: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) 1e-8)
   (- (* PI l) (/ PI (* F (/ F l))))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= 1e-8) {
		tmp = (((double) M_PI) * l) - (((double) M_PI) / (F * (F / l)));
	} else {
		tmp = (((double) M_PI) * l) - (tan((((double) M_PI) * l)) / (F * F));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((Math.PI * l) <= 1e-8) {
		tmp = (Math.PI * l) - (Math.PI / (F * (F / l)));
	} else {
		tmp = (Math.PI * l) - (Math.tan((Math.PI * l)) / (F * F));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (math.pi * l) <= 1e-8:
		tmp = (math.pi * l) - (math.pi / (F * (F / l)))
	else:
		tmp = (math.pi * l) - (math.tan((math.pi * l)) / (F * F))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(pi * l) <= 1e-8)
		tmp = Float64(Float64(pi * l) - Float64(pi / Float64(F * Float64(F / l))));
	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 ((pi * l) <= 1e-8)
		tmp = (pi * l) - (pi / (F * (F / l)));
	else
		tmp = (pi * l) - (tan((pi * l)) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], 1e-8], N[(N[(Pi * l), $MachinePrecision] - N[(Pi / N[(F * N[(F / l), $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 (*.f64 (PI.f64) l) < 1e-8

   1. Initial program 86.1%

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

      1. *-commutative 86.1%

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

      2. sqr-neg 86.1%

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

      3. associate-*r/ 86.8%

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

      4. sqr-neg 86.8%

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

      5. *-rgt-identity 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in l around 0 81.2%

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

      1. *-commutative 81.2%

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

      2. times-frac 86.4%

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

   7. Applied egg-rr 86.4%

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

      1. *-commutative 86.4%

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

      2. clear-num 86.4%

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

      3. frac-times 86.4%

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

      4. *-un-lft-identity 86.4%

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

   9. Applied egg-rr 86.4%

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

   if 1e-8 < (*.f64 (PI.f64) l)

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

      2. sqr-neg 68.6%

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

      3. associate-*r/ 68.6%

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

      4. sqr-neg 68.6%

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

      5. *-rgt-identity 68.6%

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

   3. Simplified 68.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 10^{-8}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% 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 81.2%

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

   1. associate-*l/ 81.7%

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

   2. *-un-lft-identity 81.7%

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

   3. associate-/r* 85.5%

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

4. Applied egg-rr 85.5%

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

Alternative 9: 81.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. associate-*l/ 81.7%

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

   2. *-un-lft-identity 81.7%

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

   3. associate-/r* 85.5%

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

4. Applied egg-rr 85.5%

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

   1. tan-quot 85.5%

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

6. Applied egg-rr 85.5%

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

7. Taylor expanded in l around 0 84.5%

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

8. Taylor expanded in l around inf 84.5%

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

9. Final simplification 84.5%

   \[\leadsto \pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F} \]
10. Add Preprocessing

Alternative 10: 74.9% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. associate-*l/ 81.7%

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

   2. *-un-lft-identity 81.7%

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

   3. associate-/r* 85.5%

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

4. Applied egg-rr 85.5%

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

5. Taylor expanded in l around 0 77.3%

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

6. Final simplification 77.3%

   \[\leadsto \pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F} \]
7. Add Preprocessing

Alternative 11: 75.0% accurate, 10.3× 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 81.2%

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

   1. *-commutative 81.2%

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

   2. sqr-neg 81.2%

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

   3. associate-*r/ 81.7%

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

   4. sqr-neg 81.7%

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

   5. *-rgt-identity 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in l around 0 73.5%

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

   1. *-commutative 73.5%

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

   2. times-frac 77.3%

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

7. Applied egg-rr 77.3%

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

   1. *-commutative 77.3%

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

   2. clear-num 77.3%

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

   3. frac-times 77.3%

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

   4. *-un-lft-identity 77.3%

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

9. Applied egg-rr 77.3%

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

10. Final simplification 77.3%

    \[\leadsto \pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}} \]
11. Add Preprocessing

Alternative 12: 75.0% accurate, 10.3× 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 81.2%

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

   1. *-commutative 81.2%

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

   2. sqr-neg 81.2%

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

   3. associate-*r/ 81.7%

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

   4. sqr-neg 81.7%

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

   5. *-rgt-identity 81.7%

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

3. Simplified 81.7%

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

5. Taylor expanded in l around 0 73.5%

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

   1. *-commutative 73.5%

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

   2. times-frac 77.3%

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

7. Applied egg-rr 77.3%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))