VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.3% → 96.9%

Time: 22.1s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. F = -5.4886873784888575e+124
  2. l = 232812269402098.5

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: 96.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell + \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{-1 - {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (+
  (* PI l)
  (/
   (/
    (/ (sin (* PI l)) F)
    (-
     -1.0
     (*
      (pow l 2.0)
      (+
       (* -0.5 (pow PI 2.0))
       (*
        (pow l 2.0)
        (+
         (* -0.001388888888888889 (* (pow l 2.0) (pow PI 6.0)))
         (* 0.041666666666666664 (pow PI 4.0))))))))
   F)))

C

double code(double F, double l) {
	return (((double) M_PI) * l) + (((sin((((double) M_PI) * l)) / F) / (-1.0 - (pow(l, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (pow(l, 2.0) * ((-0.001388888888888889 * (pow(l, 2.0) * pow(((double) M_PI), 6.0))) + (0.041666666666666664 * pow(((double) M_PI), 4.0)))))))) / F);
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) + (((Math.sin((Math.PI * l)) / F) / (-1.0 - (Math.pow(l, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (Math.pow(l, 2.0) * ((-0.001388888888888889 * (Math.pow(l, 2.0) * Math.pow(Math.PI, 6.0))) + (0.041666666666666664 * Math.pow(Math.PI, 4.0)))))))) / F);
}

Python

def code(F, l):
	return (math.pi * l) + (((math.sin((math.pi * l)) / F) / (-1.0 - (math.pow(l, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (math.pow(l, 2.0) * ((-0.001388888888888889 * (math.pow(l, 2.0) * math.pow(math.pi, 6.0))) + (0.041666666666666664 * math.pow(math.pi, 4.0)))))))) / F)

Julia

function code(F, l)
	return Float64(Float64(pi * l) + Float64(Float64(Float64(sin(Float64(pi * l)) / F) / Float64(-1.0 - Float64((l ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64((l ^ 2.0) * Float64(Float64(-0.001388888888888889 * Float64((l ^ 2.0) * (pi ^ 6.0))) + Float64(0.041666666666666664 * (pi ^ 4.0)))))))) / F))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) + (((sin((pi * l)) / F) / (-1.0 - ((l ^ 2.0) * ((-0.5 * (pi ^ 2.0)) + ((l ^ 2.0) * ((-0.001388888888888889 * ((l ^ 2.0) * (pi ^ 6.0))) + (0.041666666666666664 * (pi ^ 4.0)))))))) / F);
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[(N[Sin[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / N[(-1.0 - N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(-0.001388888888888889 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.6%

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

   1. associate-*l/ 74.8%

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

   2. *-un-lft-identity 74.8%

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

   3. associate-/r* 81.5%

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

4. Applied egg-rr 81.5%

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

   1. add-cube-cbrt 81.1%

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

   2. pow3 81.0%

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

6. Applied egg-rr 81.0%

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

7. Taylor expanded in l around inf 81.5%

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

   1. associate-/r* 81.5%

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

9. Simplified 81.5%

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

10. Taylor expanded in l around 0 95.4%

    \[\leadsto \pi \cdot \ell - \frac{\frac{\frac{\sin \left(\ell \cdot \pi\right)}{F}}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}}{F} \]

11. Final simplification 95.4%

    \[\leadsto \pi \cdot \ell + \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{-1 - {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + {\ell}^{2} \cdot \left(-0.001388888888888889 \cdot \left({\ell}^{2} \cdot {\pi}^{6}\right) + 0.041666666666666664 \cdot {\pi}^{4}\right)\right)}}{F} \]
12. Add Preprocessing

Alternative 2: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell + \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{-1 - {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + 0.041666666666666664 \cdot \left({\ell}^{2} \cdot {\pi}^{4}\right)\right)}}{F} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (+
  (* PI l)
  (/
   (/
    (/ (sin (* PI l)) F)
    (-
     -1.0
     (*
      (pow l 2.0)
      (+
       (* -0.5 (pow PI 2.0))
       (* 0.041666666666666664 (* (pow l 2.0) (pow PI 4.0)))))))
   F)))

C

double code(double F, double l) {
	return (((double) M_PI) * l) + (((sin((((double) M_PI) * l)) / F) / (-1.0 - (pow(l, 2.0) * ((-0.5 * pow(((double) M_PI), 2.0)) + (0.041666666666666664 * (pow(l, 2.0) * pow(((double) M_PI), 4.0))))))) / F);
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) + (((Math.sin((Math.PI * l)) / F) / (-1.0 - (Math.pow(l, 2.0) * ((-0.5 * Math.pow(Math.PI, 2.0)) + (0.041666666666666664 * (Math.pow(l, 2.0) * Math.pow(Math.PI, 4.0))))))) / F);
}

Python

def code(F, l):
	return (math.pi * l) + (((math.sin((math.pi * l)) / F) / (-1.0 - (math.pow(l, 2.0) * ((-0.5 * math.pow(math.pi, 2.0)) + (0.041666666666666664 * (math.pow(l, 2.0) * math.pow(math.pi, 4.0))))))) / F)

Julia

function code(F, l)
	return Float64(Float64(pi * l) + Float64(Float64(Float64(sin(Float64(pi * l)) / F) / Float64(-1.0 - Float64((l ^ 2.0) * Float64(Float64(-0.5 * (pi ^ 2.0)) + Float64(0.041666666666666664 * Float64((l ^ 2.0) * (pi ^ 4.0))))))) / F))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) + (((sin((pi * l)) / F) / (-1.0 - ((l ^ 2.0) * ((-0.5 * (pi ^ 2.0)) + (0.041666666666666664 * ((l ^ 2.0) * (pi ^ 4.0))))))) / F);
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[(N[Sin[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / N[(-1.0 - N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(-0.5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.041666666666666664 * N[(N[Power[l, 2.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.6%

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

   1. associate-*l/ 74.8%

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

   2. *-un-lft-identity 74.8%

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

   3. associate-/r* 81.5%

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

4. Applied egg-rr 81.5%

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

   1. add-cube-cbrt 81.1%

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

   2. pow3 81.0%

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

6. Applied egg-rr 81.0%

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

7. Taylor expanded in l around inf 81.5%

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

   1. associate-/r* 81.5%

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

9. Simplified 81.5%

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

10. Taylor expanded in l around 0 93.9%

    \[\leadsto \pi \cdot \ell - \frac{\frac{\frac{\sin \left(\ell \cdot \pi\right)}{F}}{\color{blue}{1 + {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + 0.041666666666666664 \cdot \left({\ell}^{2} \cdot {\pi}^{4}\right)\right)}}}{F} \]

11. Final simplification 93.9%

    \[\leadsto \pi \cdot \ell + \frac{\frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{-1 - {\ell}^{2} \cdot \left(-0.5 \cdot {\pi}^{2} + 0.041666666666666664 \cdot \left({\ell}^{2} \cdot {\pi}^{4}\right)\right)}}{F} \]
12. Add Preprocessing

Alternative 3: 92.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

   1. associate-*l/ 74.8%

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

   2. *-un-lft-identity 74.8%

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

   3. associate-/r* 81.5%

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

4. Applied egg-rr 81.5%

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

   1. add-cube-cbrt 81.1%

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

   2. pow3 81.0%

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

6. Applied egg-rr 81.0%

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

7. Taylor expanded in l around inf 81.5%

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

   1. associate-/r* 81.5%

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

9. Simplified 81.5%

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

10. Taylor expanded in l around 0 91.6%

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

    1. +-commutative 91.6%

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

    2. fma-define 91.6%

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

    3. unpow2 91.6%

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

    4. unpow2 91.6%

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

    5. swap-sqr 91.6%

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

    6. unpow2 91.6%

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

12. Simplified 91.6%

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

13. Final simplification 91.6%

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

Alternative 4: 65.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{\frac{\tan \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \ell\right)\right)\right)}{F}}{F} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (/ (/ (tan (expm1 (log1p (* PI l)))) F) F)))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((tan(expm1(log1p((((double) M_PI) * l)))) / F) / F);
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((Math.tan(Math.expm1(Math.log1p((Math.PI * l)))) / F) / F);
}

Python

def code(F, l):
	return (math.pi * l) - ((math.tan(math.expm1(math.log1p((math.pi * l)))) / F) / F)

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(tan(expm1(log1p(Float64(pi * l)))) / F) / F))
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Exp[N[Log[1 + N[(Pi * l), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.6%

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

   1. associate-*l/ 74.8%

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

   2. *-un-lft-identity 74.8%

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

   3. associate-/r* 81.5%

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

4. Applied egg-rr 81.5%

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

   1. expm1-log1p-u 69.6%

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

6. Applied egg-rr 69.6%

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

Alternative 5: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 3 \cdot 10^{-22}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) 3e-22)
   (- (* PI l) (/ (/ (* PI l) F) F))
   (- (* PI l) (/ (tan (* PI l)) (* F F)))))

C

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= 3e-22) {
		tmp = (((double) M_PI) * l) - (((((double) M_PI) * l) / F) / 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 ((Math.PI * l) <= 3e-22) {
		tmp = (Math.PI * l) - (((Math.PI * l) / F) / F);
	} 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) <= 3e-22:
		tmp = (math.pi * l) - (((math.pi * l) / F) / F)
	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) <= 3e-22)
		tmp = Float64(Float64(pi * l) - Float64(Float64(Float64(pi * l) / F) / 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 ((pi * l) <= 3e-22)
		tmp = (pi * l) - (((pi * l) / F) / F);
	else
		tmp = (pi * l) - (tan((pi * l)) / (F * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], 3e-22], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[(Pi * l), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < 2.9999999999999999e-22

   1. Initial program 78.4%

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

      1. associate-*l/ 78.8%

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

      2. *-un-lft-identity 78.8%

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

      3. associate-/r* 88.5%

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

   4. Applied egg-rr 88.5%

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

   5. Taylor expanded in l around 0 82.4%

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

   if 2.9999999999999999e-22 < (*.f64 (PI.f64) l)

   1. Initial program 66.2%

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

      1. *-commutative 66.2%

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

      2. sqr-neg 66.2%

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

      3. associate-*r/ 66.2%

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

      4. sqr-neg 66.2%

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

      5. *-rgt-identity 66.2%

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

   3. Simplified 66.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 3 \cdot 10^{-22}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 74.6%

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

   1. associate-*l/ 74.8%

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

   2. *-un-lft-identity 74.8%

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

   3. associate-/r* 81.5%

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

4. Applied egg-rr 81.5%

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

Alternative 7: 74.8% 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 74.6%

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

   1. associate-*l/ 74.8%

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

   2. *-un-lft-identity 74.8%

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

   3. associate-/r* 81.5%

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

4. Applied egg-rr 81.5%

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

5. Taylor expanded in l around 0 73.6%

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

6. Final simplification 73.6%

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

Alternative 8: 74.9% 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 74.6%

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

   1. *-commutative 74.6%

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

   2. sqr-neg 74.6%

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

   3. associate-*r/ 74.8%

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

   4. sqr-neg 74.8%

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

   5. *-rgt-identity 74.8%

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

3. Simplified 74.8%

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

5. Taylor expanded in l around 0 66.9%

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

   1. *-commutative 66.9%

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

   2. times-frac 73.6%

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

7. Applied egg-rr 73.6%

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

Reproduce

herbie shell --seed 2024113 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))