VandenBroeck and Keller, Equation (6)

Percentage Accurate: 76.6% → 82.5%

Time: 16.5s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. F = -8.249765863705203e-266
  2. l = -4128032183775673.5

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: 76.6% 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: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \pi \cdot \ell - \frac{\frac{\tan \left(\ell \cdot \sqrt[3]{{\pi}^{3}}\right)}{F}}{F} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (/ (/ (tan (* l (cbrt (pow PI 3.0)))) F) F)))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((tan((l * cbrt(pow(((double) M_PI), 3.0)))) / F) / F);
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((Math.tan((l * Math.cbrt(Math.pow(Math.PI, 3.0)))) / F) / F);
}

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(tan(Float64(l * cbrt((pi ^ 3.0)))) / F) / F))
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(l * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.6%

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

   1. associate-*l/ 78.1%

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

   2. *-un-lft-identity 78.1%

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

   3. associate-/r* 84.5%

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

3. Applied egg-rr 84.5%

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

   1. add-cbrt-cube 84.6%

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

   2. pow3 84.6%

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

5. Applied egg-rr 84.6%

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

6. Final simplification 84.6%

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

Alternative 2: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{F \cdot F} \leq 10^{+299}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{F}}{\frac{F}{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (if (<= (/ 1.0 (* F F)) 1e+299)
   (- (* PI l) (/ (tan (* PI l)) (* F F)))
   (- (* PI l) (/ (/ PI F) (/ F l)))))

C

double code(double F, double l) {
	double tmp;
	if ((1.0 / (F * F)) <= 1e+299) {
		tmp = (((double) M_PI) * l) - (tan((((double) M_PI) * l)) / (F * F));
	} else {
		tmp = (((double) M_PI) * l) - ((((double) M_PI) / F) / (F / l));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	double tmp;
	if ((1.0 / (F * F)) <= 1e+299) {
		tmp = (Math.PI * l) - (Math.tan((Math.PI * l)) / (F * F));
	} else {
		tmp = (Math.PI * l) - ((Math.PI / F) / (F / l));
	}
	return tmp;
}

Python

def code(F, l):
	tmp = 0
	if (1.0 / (F * F)) <= 1e+299:
		tmp = (math.pi * l) - (math.tan((math.pi * l)) / (F * F))
	else:
		tmp = (math.pi * l) - ((math.pi / F) / (F / l))
	return tmp

Julia

function code(F, l)
	tmp = 0.0
	if (Float64(1.0 / Float64(F * F)) <= 1e+299)
		tmp = Float64(Float64(pi * l) - Float64(tan(Float64(pi * l)) / Float64(F * F)));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(pi / F) / Float64(F / l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, l)
	tmp = 0.0;
	if ((1.0 / (F * F)) <= 1e+299)
		tmp = (pi * l) - (tan((pi * l)) / (F * F));
	else
		tmp = (pi * l) - ((pi / F) / (F / l));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, l_] := If[LessEqual[N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision], 1e+299], N[(N[(Pi * l), $MachinePrecision] - N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(Pi / F), $MachinePrecision] / N[(F / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 1 (*.f64 F F)) < 1.0000000000000001e299

   1. Initial program 93.3%

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

      1. sqr-neg 93.3%

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

      2. associate-*l/ 93.3%

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

      3. *-lft-identity 93.3%

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

      4. sqr-neg 93.3%

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

   3. Simplified 93.3%

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

   if 1.0000000000000001e299 < (/.f64 1 (*.f64 F F))

   1. Initial program 23.8%

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

   2. Taylor expanded in l around 0 23.3%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 23.2%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]

      2. associate-/r/ 23.3%

         \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{{F}^{2}} \cdot \pi} \]

      3. unpow2 23.3%

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

   4. Simplified 23.3%

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

      1. *-commutative 23.3%

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

      2. clear-num 23.3%

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

      3. un-div-inv 23.3%

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

      4. *-un-lft-identity 23.3%

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

      5. times-frac 51.6%

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

      6. /-rgt-identity 51.6%

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

   6. Applied egg-rr 51.6%

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

      1. associate-/r* 51.7%

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

      2. div-inv 51.7%

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

      3. clear-num 51.7%

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

   8. Applied egg-rr 51.7%

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

      1. clear-num 51.7%

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

      2. un-div-inv 51.7%

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

   10. Applied egg-rr 51.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{F \cdot F} \leq 10^{+299}:\\ \;\;\;\;\pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi}{F}}{\frac{F}{\ell}}\\ \end{array} \]

Alternative 3: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

   1. sqr-neg 77.6%

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

   2. associate-*l/ 78.1%

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

   3. *-lft-identity 78.1%

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

   4. sqr-neg 78.1%

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

3. Simplified 78.1%

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

   1. associate-/r* 84.5%

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

   2. div-inv 84.5%

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

5. Applied egg-rr 84.5%

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

6. Final simplification 84.5%

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

Alternative 4: 82.5% 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 77.6%

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

   1. associate-*l/ 78.1%

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

   2. *-un-lft-identity 78.1%

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

   3. associate-/r* 84.5%

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

3. Applied egg-rr 84.5%

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

4. Final simplification 84.5%

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

Alternative 5: 69.5% accurate, 1.5× speedup

Math

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

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

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in l around 0 71.9%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
3. Step-by-step derivation

   1. associate-/l* 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]

   2. associate-/r/ 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{{F}^{2}} \cdot \pi} \]

   3. unpow2 71.9%

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

4. Simplified 71.9%

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

5. Final simplification 71.9%

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

Alternative 6: 74.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in l around 0 71.9%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
3. Step-by-step derivation

   1. associate-/l* 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]

   2. associate-/r/ 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{{F}^{2}} \cdot \pi} \]

   3. unpow2 71.9%

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

4. Simplified 71.9%

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

   1. *-commutative 71.9%

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

   2. clear-num 71.9%

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

   3. un-div-inv 71.9%

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

   4. *-un-lft-identity 71.9%

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

   5. times-frac 78.3%

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

   6. /-rgt-identity 78.3%

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

6. Applied egg-rr 78.3%

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

   1. associate-/r* 78.4%

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

   2. div-inv 78.3%

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

   3. clear-num 78.3%

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

8. Applied egg-rr 78.3%

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

9. Final simplification 78.3%

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

Alternative 7: 74.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in l around 0 71.9%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
3. Step-by-step derivation

   1. associate-/l* 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]

   2. associate-/r/ 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{{F}^{2}} \cdot \pi} \]

   3. unpow2 71.9%

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

4. Simplified 71.9%

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

   1. *-commutative 71.9%

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

   2. clear-num 71.9%

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

   3. un-div-inv 71.9%

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

   4. *-un-lft-identity 71.9%

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

   5. times-frac 78.3%

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

   6. /-rgt-identity 78.3%

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

6. Applied egg-rr 78.3%

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

7. Final simplification 78.3%

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

Alternative 8: 74.9% accurate, 1.5× 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 77.6%

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

2. Taylor expanded in l around 0 71.9%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
3. Step-by-step derivation

   1. associate-/l* 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]

   2. associate-/r/ 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{{F}^{2}} \cdot \pi} \]

   3. unpow2 71.9%

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

4. Simplified 71.9%

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

   1. *-commutative 71.9%

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

   2. clear-num 71.9%

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

   3. un-div-inv 71.9%

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

   4. *-un-lft-identity 71.9%

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

   5. times-frac 78.3%

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

   6. /-rgt-identity 78.3%

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

6. Applied egg-rr 78.3%

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

   1. associate-/r* 78.4%

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

   2. div-inv 78.3%

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

   3. clear-num 78.3%

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

8. Applied egg-rr 78.3%

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

   1. associate-*r/ 78.4%

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

10. Applied egg-rr 78.4%

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

11. Final simplification 78.4%

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

Alternative 9: 74.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

2. Taylor expanded in l around 0 71.9%

   \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]
3. Step-by-step derivation

   1. associate-/l* 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]

   2. associate-/r/ 71.9%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{{F}^{2}} \cdot \pi} \]

   3. unpow2 71.9%

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

4. Simplified 71.9%

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

   1. *-commutative 71.9%

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

   2. clear-num 71.9%

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

   3. un-div-inv 71.9%

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

   4. *-un-lft-identity 71.9%

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

   5. times-frac 78.3%

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

   6. /-rgt-identity 78.3%

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

6. Applied egg-rr 78.3%

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

   1. associate-/r* 78.4%

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

   2. div-inv 78.3%

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

   3. clear-num 78.3%

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

8. Applied egg-rr 78.3%

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

   1. *-commutative 78.3%

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

   2. clear-num 78.3%

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

   3. un-div-inv 78.4%

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

10. Applied egg-rr 78.4%

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

11. Final simplification 78.4%

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

Reproduce

herbie shell --seed 2023292 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))