VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.4% → 96.6%

Time: 25.9s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. F = -2.3830813878912138e-33
  2. l = -3321791691070032.5

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: 76.4% 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: 96.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate-*l/ 76.7%

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

   2. *-un-lft-identity 76.7%

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

   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. tan-quot 81.5%

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

   2. clear-num 81.5%

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

6. Applied egg-rr 81.5%

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

7. Taylor expanded in l around 0 95.7%

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

8. Final simplification 95.7%

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

Alternative 2: 95.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate-*l/ 76.7%

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

   2. *-un-lft-identity 76.7%

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

   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. tan-quot 81.5%

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

   2. clear-num 81.5%

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

6. Applied egg-rr 81.5%

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

7. Taylor expanded in l around 0 95.4%

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

8. Final simplification 95.4%

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

Alternative 3: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{\mathsf{fma}\left({\ell}^{2}, \pi \cdot 0.16666666666666666 - {\ell}^{2} \cdot \left({\pi}^{3} \cdot -0.019444444444444445\right), \frac{1}{\pi}\right)}{\ell}} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (+
  (* PI l)
  (/
   (/ -1.0 F)
   (*
    F
    (/
     (fma
      (pow l 2.0)
      (-
       (* PI 0.16666666666666666)
       (* (pow l 2.0) (* (pow PI 3.0) -0.019444444444444445)))
      (/ 1.0 PI))
     l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) + ((-1.0 / F) / (F * (fma(pow(l, 2.0), ((((double) M_PI) * 0.16666666666666666) - (pow(l, 2.0) * (pow(((double) M_PI), 3.0) * -0.019444444444444445))), (1.0 / ((double) M_PI))) / l)));
}

Julia

function code(F, l)
	return Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F * Float64(fma((l ^ 2.0), Float64(Float64(pi * 0.16666666666666666) - Float64((l ^ 2.0) * Float64((pi ^ 3.0) * -0.019444444444444445))), Float64(1.0 / pi)) / l))))
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F * N[(N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(Pi * 0.16666666666666666), $MachinePrecision] - N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.019444444444444445), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

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

   1. tan-quot 76.4%

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

4. Applied egg-rr 76.4%

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

5. Taylor expanded in l around 0 75.6%

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

   1. /-rgt-identity 75.6%

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

   2. *-commutative 75.6%

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

   3. /-rgt-identity 75.6%

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

   4. clear-num 75.6%

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

   5. associate-/r* 75.5%

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

   6. metadata-eval 75.5%

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

   7. add-sqr-sqrt 36.8%

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

   8. sqrt-prod 64.6%

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

   9. sqrt-div 64.6%

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

   10. frac-times 64.6%

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

   11. *-un-lft-identity 64.6%

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

   12. sqrt-div 64.9%

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

   13. metadata-eval 64.9%

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

   14. sqrt-prod 39.7%

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

   15. add-sqr-sqrt 80.6%

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

7. Applied egg-rr 80.6%

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

8. Taylor expanded in l around 0 95.0%

   \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\frac{{\ell}^{2} \cdot \left(-1 \cdot \left({\ell}^{2} \cdot \left(-0.027777777777777776 \cdot {\pi}^{3} + 0.008333333333333333 \cdot {\pi}^{3}\right)\right) - -0.16666666666666666 \cdot \pi\right) + \frac{1}{\pi}}{\ell}} \cdot F} \]
9. Step-by-step derivation

   1. fma-define 95.0%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\color{blue}{\mathsf{fma}\left({\ell}^{2}, -1 \cdot \left({\ell}^{2} \cdot \left(-0.027777777777777776 \cdot {\pi}^{3} + 0.008333333333333333 \cdot {\pi}^{3}\right)\right) - -0.16666666666666666 \cdot \pi, \frac{1}{\pi}\right)}}{\ell} \cdot F} \]

   2. cancel-sign-sub-inv 95.0%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left({\ell}^{2}, \color{blue}{-1 \cdot \left({\ell}^{2} \cdot \left(-0.027777777777777776 \cdot {\pi}^{3} + 0.008333333333333333 \cdot {\pi}^{3}\right)\right) + \left(--0.16666666666666666\right) \cdot \pi}, \frac{1}{\pi}\right)}{\ell} \cdot F} \]

   3. mul-1-neg 95.0%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left({\ell}^{2}, \color{blue}{\left(-{\ell}^{2} \cdot \left(-0.027777777777777776 \cdot {\pi}^{3} + 0.008333333333333333 \cdot {\pi}^{3}\right)\right)} + \left(--0.16666666666666666\right) \cdot \pi, \frac{1}{\pi}\right)}{\ell} \cdot F} \]

   4. distribute-rgt-out 95.0%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left({\ell}^{2}, \left(-{\ell}^{2} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(-0.027777777777777776 + 0.008333333333333333\right)\right)}\right) + \left(--0.16666666666666666\right) \cdot \pi, \frac{1}{\pi}\right)}{\ell} \cdot F} \]

   5. metadata-eval 95.0%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left({\ell}^{2}, \left(-{\ell}^{2} \cdot \left({\pi}^{3} \cdot \color{blue}{-0.019444444444444445}\right)\right) + \left(--0.16666666666666666\right) \cdot \pi, \frac{1}{\pi}\right)}{\ell} \cdot F} \]

   6. metadata-eval 95.0%

      \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\frac{\mathsf{fma}\left({\ell}^{2}, \left(-{\ell}^{2} \cdot \left({\pi}^{3} \cdot -0.019444444444444445\right)\right) + \color{blue}{0.16666666666666666} \cdot \pi, \frac{1}{\pi}\right)}{\ell} \cdot F} \]

10. Simplified 95.0%

    \[\leadsto \pi \cdot \ell - \frac{\frac{1}{F}}{\color{blue}{\frac{\mathsf{fma}\left({\ell}^{2}, \left(-{\ell}^{2} \cdot \left({\pi}^{3} \cdot -0.019444444444444445\right)\right) + 0.16666666666666666 \cdot \pi, \frac{1}{\pi}\right)}{\ell}} \cdot F} \]

11. Final simplification 95.0%

    \[\leadsto \pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{\mathsf{fma}\left({\ell}^{2}, \pi \cdot 0.16666666666666666 - {\ell}^{2} \cdot \left({\pi}^{3} \cdot -0.019444444444444445\right), \frac{1}{\pi}\right)}{\ell}} \]
12. Add Preprocessing

Alternative 4: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \cdot F \leq 10^{-308}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{\frac{1}{\pi} + 0.16666666666666666 \cdot \left(\pi \cdot {\ell}^{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)} \cdot \frac{-1}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* F F) 1e-308)
   (+
    (* PI l)
    (/
     (/ -1.0 F)
     (* F (/ (+ (/ 1.0 PI) (* 0.16666666666666666 (* PI (pow l 2.0)))) l))))
   (+
    (* PI l)
    (*
     (/ (sin (* PI l)) (fma -0.5 (pow (* PI l) 2.0) 1.0))
     (/ -1.0 (* F F))))))

C

double code(double F, double l) {
	double tmp;
	if ((F * F) <= 1e-308) {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F * (((1.0 / ((double) M_PI)) + (0.16666666666666666 * (((double) M_PI) * pow(l, 2.0)))) / l)));
	} else {
		tmp = (((double) M_PI) * l) + ((sin((((double) M_PI) * l)) / fma(-0.5, pow((((double) M_PI) * l), 2.0), 1.0)) * (-1.0 / (F * F)));
	}
	return tmp;
}

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(F * F) <= 1e-308)
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F * Float64(Float64(Float64(1.0 / pi) + Float64(0.16666666666666666 * Float64(pi * (l ^ 2.0)))) / l))));
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(sin(Float64(pi * l)) / fma(-0.5, (Float64(pi * l) ^ 2.0), 1.0)) * Float64(-1.0 / Float64(F * F))));
	end
	return tmp
end

Wolfram

code[F_, l_] := If[LessEqual[N[(F * F), $MachinePrecision], 1e-308], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F * N[(N[(N[(1.0 / Pi), $MachinePrecision] + N[(0.16666666666666666 * N[(Pi * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(N[Sin[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(-0.5 * N[Power[N[(Pi * l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 F F) < 9.9999999999999991e-309

   1. Initial program 37.3%

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

      1. tan-quot 37.3%

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

   4. Applied egg-rr 37.3%

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

   5. Taylor expanded in l around 0 35.7%

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

      1. /-rgt-identity 35.7%

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

      2. *-commutative 35.7%

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

      3. /-rgt-identity 35.7%

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

      4. clear-num 35.7%

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

      5. associate-/r* 35.7%

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

      6. metadata-eval 35.7%

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

      7. add-sqr-sqrt 16.9%

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

      8. sqrt-prod 17.9%

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

      9. sqrt-div 17.9%

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

      10. frac-times 18.0%

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

      11. *-un-lft-identity 18.0%

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

      12. sqrt-div 19.2%

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

      13. metadata-eval 19.2%

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

      14. sqrt-prod 27.8%

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

      15. add-sqr-sqrt 54.5%

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

   7. Applied egg-rr 54.5%

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

   8. Taylor expanded in l around 0 72.4%

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

   if 9.9999999999999991e-309 < (*.f64 F F)

   1. Initial program 90.8%

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

      1. tan-quot 90.8%

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in l around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. fma-define 97.9%

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

      3. *-commutative 97.9%

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

      4. unpow2 97.9%

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

      5. unpow2 97.9%

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

      6. swap-sqr 97.9%

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

      7. unpow2 97.9%

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

      8. *-commutative 97.9%

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

   7. Simplified 97.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \cdot F \leq 10^{-308}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{\frac{1}{\pi} + 0.16666666666666666 \cdot \left(\pi \cdot {\ell}^{2}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\sin \left(\pi \cdot \ell\right)}{\mathsf{fma}\left(-0.5, {\left(\pi \cdot \ell\right)}^{2}, 1\right)} \cdot \frac{-1}{F \cdot F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (F l)
 :precision binary64
 (+
  (* PI l)
  (/ (/ (/ -1.0 (/ (fma -0.5 (pow (* PI l) 2.0) 1.0) (sin (* PI l)))) F) 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) / sin((((double) M_PI) * l)))) / F) / F);
}

Julia

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate-*l/ 76.7%

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

   2. *-un-lft-identity 76.7%

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

   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. tan-quot 81.5%

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

   2. clear-num 81.5%

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

6. Applied egg-rr 81.5%

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

7. Taylor expanded in l around 0 91.9%

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

   1. +-commutative 91.9%

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

   2. fma-define 91.9%

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

   3. *-commutative 91.9%

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

   4. unpow2 91.9%

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

   5. unpow2 91.9%

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

   6. swap-sqr 91.9%

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

   7. unpow2 91.9%

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

   8. *-commutative 91.9%

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

9. Simplified 91.9%

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

10. Final simplification 91.9%

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

Alternative 6: 90.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{\frac{1}{\pi} + 0.16666666666666666 \cdot \left(\pi \cdot {\ell}^{2}\right)}{\ell}} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (+
  (* PI l)
  (/
   (/ -1.0 F)
   (* F (/ (+ (/ 1.0 PI) (* 0.16666666666666666 (* PI (pow l 2.0)))) l)))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) + ((-1.0 / F) / (F * (((1.0 / ((double) M_PI)) + (0.16666666666666666 * (((double) M_PI) * pow(l, 2.0)))) / l)));
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) + ((-1.0 / F) / (F * (((1.0 / Math.PI) + (0.16666666666666666 * (Math.PI * Math.pow(l, 2.0)))) / l)));
}

Python

def code(F, l):
	return (math.pi * l) + ((-1.0 / F) / (F * (((1.0 / math.pi) + (0.16666666666666666 * (math.pi * math.pow(l, 2.0)))) / l)))

Julia

function code(F, l)
	return Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F * Float64(Float64(Float64(1.0 / pi) + Float64(0.16666666666666666 * Float64(pi * (l ^ 2.0)))) / l))))
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) + ((-1.0 / F) / (F * (((1.0 / pi) + (0.16666666666666666 * (pi * (l ^ 2.0)))) / l)));
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F * N[(N[(N[(1.0 / Pi), $MachinePrecision] + N[(0.16666666666666666 * N[(Pi * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.4%

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

   1. tan-quot 76.4%

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

4. Applied egg-rr 76.4%

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

5. Taylor expanded in l around 0 75.6%

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

   1. /-rgt-identity 75.6%

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

   2. *-commutative 75.6%

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

   3. /-rgt-identity 75.6%

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

   4. clear-num 75.6%

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

   5. associate-/r* 75.5%

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

   6. metadata-eval 75.5%

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

   7. add-sqr-sqrt 36.8%

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

   8. sqrt-prod 64.6%

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

   9. sqrt-div 64.6%

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

   10. frac-times 64.6%

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

   11. *-un-lft-identity 64.6%

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

   12. sqrt-div 64.9%

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

   13. metadata-eval 64.9%

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

   14. sqrt-prod 39.7%

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

   15. add-sqr-sqrt 80.6%

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

7. Applied egg-rr 80.6%

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

8. Taylor expanded in l around 0 88.2%

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

9. Final simplification 88.2%

   \[\leadsto \pi \cdot \ell + \frac{\frac{-1}{F}}{F \cdot \frac{\frac{1}{\pi} + 0.16666666666666666 \cdot \left(\pi \cdot {\ell}^{2}\right)}{\ell}} \]
10. Add Preprocessing

Alternative 7: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{F}}{\frac{F}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) 2e-77)
   (- (* 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) <= 2e-77) {
		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) <= 2e-77) {
		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) <= 2e-77:
		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) <= 2e-77)
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / 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) <= 2e-77)
		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], 2e-77], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] / N[(F / l), $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) < 1.9999999999999999e-77

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. sqr-neg 79.8%

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

      3. associate-*r/ 80.2%

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

      4. sqr-neg 80.2%

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

      5. *-rgt-identity 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in l around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. times-frac 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. clear-num 83.5%

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

      2. un-div-inv 83.6%

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

   9. Applied egg-rr 83.6%

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

   if 1.9999999999999999e-77 < (*.f64 (PI.f64) l)

   1. Initial program 67.4%

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

      1. *-commutative 67.4%

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

      2. sqr-neg 67.4%

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

      3. associate-*r/ 67.4%

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

      4. sqr-neg 67.4%

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

      5. *-rgt-identity 67.4%

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

   3. Simplified 67.4%

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

Alternative 8: 82.6% 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 76.4%

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

   1. associate-*l/ 76.7%

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

   2. *-un-lft-identity 76.7%

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

   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 9: 81.8% 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 76.4%

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

   1. tan-quot 76.4%

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

4. Applied egg-rr 76.4%

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

5. Taylor expanded in l around 0 75.6%

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

   1. /-rgt-identity 75.6%

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

   2. associate-*l/ 75.9%

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

   3. *-un-lft-identity 75.9%

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

   4. associate-/r* 80.7%

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

7. Applied egg-rr 80.7%

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

Alternative 10: 75.2% accurate, 10.3× speedup

Math

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

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

Derivation

1. Initial program 76.4%

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

   1. *-commutative 76.4%

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

   2. sqr-neg 76.4%

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

   3. associate-*r/ 76.7%

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

   4. sqr-neg 76.7%

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

   5. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in l around 0 72.5%

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

   1. *-commutative 72.5%

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

   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. clear-num 77.3%

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

   2. un-div-inv 77.3%

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

9. Applied egg-rr 77.3%

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

Alternative 11: 75.2% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.4%

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

   1. associate-*l/ 76.7%

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

   2. *-un-lft-identity 76.7%

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

   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 77.3%

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

   1. associate-/l* 77.3%

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

7. Simplified 77.3%

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

Alternative 12: 75.2% 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 76.4%

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

   1. *-commutative 76.4%

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

   2. sqr-neg 76.4%

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

   3. associate-*r/ 76.7%

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

   4. sqr-neg 76.7%

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

   5. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in l around 0 72.5%

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

   1. *-commutative 72.5%

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

   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 13: 75.2% 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 76.4%

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

   1. *-commutative 76.4%

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

   2. sqr-neg 76.4%

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

   3. associate-*r/ 76.7%

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

   4. sqr-neg 76.7%

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

   5. *-rgt-identity 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in l around 0 72.5%

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

   1. *-commutative 72.5%

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

   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 2024101 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))