Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.0% → 80.7%

Time: 17.6s

Alternatives: 14

Speedup: 3.7×

• 26.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 55.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 80.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 2.85 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{t \cdot \frac{k}{\ell}}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.6e-63)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (if (<= k 2.85e+193)
     (/ 2.0 (* (sin k) (* (tan k) (/ (* t (* k (/ k l))) l))))
     (/ 2.0 (* (sin k) (* (tan k) (* k (/ (* t (/ k l)) l))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.6e-63) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if (k <= 2.85e+193) {
		tmp = 2.0 / (sin(k) * (tan(k) * ((t * (k * (k / l))) / l)));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5.6d-63) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else if (k <= 2.85d+193) then
        tmp = 2.0d0 / (sin(k) * (tan(k) * ((t * (k * (k / l))) / l)))
    else
        tmp = 2.0d0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.6e-63) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if (k <= 2.85e+193) {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * ((t * (k * (k / l))) / l)));
	} else {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (k * ((t * (k / l)) / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5.6e-63:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	elif k <= 2.85e+193:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * ((t * (k * (k / l))) / l)))
	else:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (k * ((t * (k / l)) / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.6e-63)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif (k <= 2.85e+193)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(Float64(t * Float64(k * Float64(k / l))) / l))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(Float64(t * Float64(k / l)) / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.6e-63)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	elseif (k <= 2.85e+193)
		tmp = 2.0 / (sin(k) * (tan(k) * ((t * (k * (k / l))) / l)));
	else
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.6e-63], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.85e+193], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.6000000000000005e-63

   1. Initial program 55.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 55.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 56.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 51.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 57.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 65.5%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 52.6%

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

       1. unpow2 52.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 52.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 57.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 65.5%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 66.0%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 5.6000000000000005e-63 < k < 2.85e193

   1. Initial program 49.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 49.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 49.5%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 49.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 49.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 49.5%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 49.5%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 49.5%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 58.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 58.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 58.1%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 70.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 70.6%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 70.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 49.9%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 25.3%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 25.0%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 25.0%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 41.5%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 58.1%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 58.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 58.1%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 58.1%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 58.1%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 70.5%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 70.5%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 72.9%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 72.9%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]
   11. Step-by-step derivation

       1. associate-*r/ 80.3%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\frac{\left(\frac{k}{\ell} \cdot k\right) \cdot t}{\ell}}\right)} \]

       2. *-commutative 80.3%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t}{\ell}\right)} \]

   12. Applied egg-rr 80.3%

       \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\frac{\left(k \cdot \frac{k}{\ell}\right) \cdot t}{\ell}}\right)} \]

   if 2.85e193 < k

   1. Initial program 37.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 37.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 37.9%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 37.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 37.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 37.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 37.9%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 37.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 62.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 62.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 62.5%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 63.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 63.3%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 63.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 18.1%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 18.1%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 17.4%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 17.4%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 17.4%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 62.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 62.5%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 62.5%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 62.5%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 62.5%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 63.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 63.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 73.4%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 73.4%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]

   11. Taylor expanded in k around 0 62.5%

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

       1. unpow2 62.5%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}\right)} \]

       2. associate-*r* 66.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}\right)} \]

       3. unpow2 66.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}\right)} \]

       4. times-frac 89.9%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)}\right)} \]

       5. associate-*r/ 86.6%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\left(k \cdot \frac{t}{\ell}\right)}\right)\right)} \]

       6. *-commutative 86.6%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(k \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}\right)} \]

       7. associate-*l* 86.6%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]

   13. Simplified 86.6%

       \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]
   14. Step-by-step derivation

       1. associate-*l/ 90.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \color{blue}{\frac{t \cdot \frac{k}{\ell}}{\ell}}\right)\right)} \]

   15. Applied egg-rr 90.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.6 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 2.85 \cdot 10^{+193}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot \frac{k}{\ell}\right)}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{t \cdot \frac{k}{\ell}}{\ell}\right)\right)}\\ \end{array} \]

Alternative 2: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right) \leq \infty:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(2 + t_1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{t \cdot \frac{k}{\ell}}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ 1.0 (+ 1.0 t_1)))
        INFINITY)
     (* l (* l (/ 2.0 (* (tan k) (* (* (pow t 3.0) (sin k)) (+ 2.0 t_1))))))
     (/ 2.0 (* (sin k) (* (tan k) (* k (/ (* t (/ k l)) l))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1))) <= ((double) INFINITY)) {
		tmp = l * (l * (2.0 / (tan(k) * ((pow(t, 3.0) * sin(k)) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1))) <= Double.POSITIVE_INFINITY) {
		tmp = l * (l * (2.0 / (Math.tan(k) * ((Math.pow(t, 3.0) * Math.sin(k)) * (2.0 + t_1)))));
	} else {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (k * ((t * (k / l)) / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + t_1))) <= math.inf:
		tmp = l * (l * (2.0 / (math.tan(k) * ((math.pow(t, 3.0) * math.sin(k)) * (2.0 + t_1)))))
	else:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (k * ((t * (k / l)) / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1))) <= Inf)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(tan(k) * Float64(Float64((t ^ 3.0) * sin(k)) * Float64(2.0 + t_1))))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(Float64(t * Float64(k / l)) / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1))) <= Inf)
		tmp = l * (l * (2.0 / (tan(k) * (((t ^ 3.0) * sin(k)) * (2.0 + t_1)))));
	else
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(l * N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[t, 3.0], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 84.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 84.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 84.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 83.6%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 85.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 85.2%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 85.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 85.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 85.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 85.2%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 85.2%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 85.2%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 67.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]

      2. expm1-udef 65.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]

      3. associate-*l* 65.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]

      4. *-commutative 65.7%

         \[\leadsto e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sin k \cdot {t}^{3}\right)}\right)}\right)\right)} - 1 \]

   5. Applied egg-rr 65.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {t}^{3}\right)\right)}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 69.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {t}^{3}\right)\right)}\right)\right)\right)} \]

      2. expm1-log1p 88.6%

         \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {t}^{3}\right)\right)}\right)} \]

      3. *-commutative 88.6%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot {t}^{3}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}\right) \]

      4. *-commutative 88.6%

         \[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left({t}^{3} \cdot \sin k\right)} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right) \]

   7. Simplified 88.6%

      \[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 0.0%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 0.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 0.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 0.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 0.0%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 37.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 37.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 37.2%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 53.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 53.3%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 53.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 37.7%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 17.8%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 11.0%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 11.0%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 23.7%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 37.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 37.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 37.1%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 37.1%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 37.1%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 53.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 53.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 62.6%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 62.6%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]

   11. Taylor expanded in k around 0 37.1%

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

       1. unpow2 37.1%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}\right)} \]

       2. associate-*r* 43.3%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}\right)} \]

       3. unpow2 43.3%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}\right)} \]

       4. times-frac 75.6%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)}\right)} \]

       5. associate-*r/ 67.8%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\left(k \cdot \frac{t}{\ell}\right)}\right)\right)} \]

       6. *-commutative 67.8%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(k \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}\right)} \]

       7. associate-*l* 64.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]

   13. Simplified 64.0%

       \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]
   14. Step-by-step derivation

       1. associate-*l/ 68.8%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \color{blue}{\frac{t \cdot \frac{k}{\ell}}{\ell}}\right)\right)} \]

   15. Applied egg-rr 68.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{t \cdot \frac{k}{\ell}}{\ell}\right)\right)}\\ \end{array} \]

Alternative 3: 79.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{t \cdot \frac{k}{\ell}}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 9.2e-64)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (if (<= k 1.6e+157)
     (/ 2.0 (* (sin k) (* (tan k) (* (* k (/ k l)) (/ t l)))))
     (/ 2.0 (* (sin k) (* (tan k) (* k (/ (* t (/ k l)) l))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.2e-64) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else if (k <= 1.6e+157) {
		tmp = 2.0 / (sin(k) * (tan(k) * ((k * (k / l)) * (t / l))));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 9.2d-64) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else if (k <= 1.6d+157) then
        tmp = 2.0d0 / (sin(k) * (tan(k) * ((k * (k / l)) * (t / l))))
    else
        tmp = 2.0d0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 9.2e-64) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else if (k <= 1.6e+157) {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * ((k * (k / l)) * (t / l))));
	} else {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (k * ((t * (k / l)) / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 9.2e-64:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	elif k <= 1.6e+157:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * ((k * (k / l)) * (t / l))))
	else:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (k * ((t * (k / l)) / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 9.2e-64)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	elseif (k <= 1.6e+157)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(Float64(k * Float64(k / l)) * Float64(t / l)))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(Float64(t * Float64(k / l)) / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 9.2e-64)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	elseif (k <= 1.6e+157)
		tmp = 2.0 / (sin(k) * (tan(k) * ((k * (k / l)) * (t / l))));
	else
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 9.2e-64], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.6e+157], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.2000000000000006e-64

   1. Initial program 55.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 55.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 56.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 51.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 57.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 65.5%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 52.6%

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

       1. unpow2 52.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 52.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 57.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 65.5%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 66.0%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 9.2000000000000006e-64 < k < 1.6e157

   1. Initial program 49.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 49.4%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 49.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 49.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 49.4%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 49.4%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 49.4%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 60.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 60.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 60.2%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 73.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 73.0%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 73.0%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 54.7%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 23.9%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 23.7%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 23.7%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 44.5%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 60.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 60.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 60.1%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 60.1%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 60.1%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 73.0%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 73.0%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 75.5%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 75.5%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]

   if 1.6e157 < k

   1. Initial program 41.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 41.0%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 41.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 41.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 41.0%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 41.0%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 41.0%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 59.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 59.3%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 59.3%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 62.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 62.7%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 62.7%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 21.3%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 21.3%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 20.6%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 20.6%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 20.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 59.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 59.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 59.3%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 59.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 59.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 62.7%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 62.7%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 70.6%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 70.6%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]

   11. Taylor expanded in k around 0 59.3%

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

       1. unpow2 59.3%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}\right)} \]

       2. associate-*r* 62.2%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}\right)} \]

       3. unpow2 62.2%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}\right)} \]

       4. times-frac 87.7%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)}\right)} \]

       5. associate-*r/ 80.4%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\left(k \cdot \frac{t}{\ell}\right)}\right)\right)} \]

       6. *-commutative 80.4%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(k \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}\right)} \]

       7. associate-*l* 80.4%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]

   13. Simplified 80.4%

       \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]
   14. Step-by-step derivation

       1. associate-*l/ 82.9%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \color{blue}{\frac{t \cdot \frac{k}{\ell}}{\ell}}\right)\right)} \]

   15. Applied egg-rr 82.9%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.2 \cdot 10^{-64}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(\left(k \cdot \frac{k}{\ell}\right) \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{t \cdot \frac{k}{\ell}}{\ell}\right)\right)}\\ \end{array} \]

Alternative 4: 72.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 2.2e-14)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (* (* l l) (/ 2.0 (* (tan k) (* (* k k) (* t (sin k))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-14) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d-14) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = (l * l) * (2.0d0 / (tan(k) * ((k * k) * (t * sin(k)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 2.2e-14) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((k * k) * (t * Math.sin(k)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 2.2e-14:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((k * k) * (t * math.sin(k)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 2.2e-14)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(k * k) * Float64(t * sin(k))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 2.2e-14)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = (l * l) * (2.0 / (tan(k) * ((k * k) * (t * sin(k)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 2.2e-14], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.2000000000000001e-14

   1. Initial program 56.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 56.1%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 56.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 56.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 56.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 56.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 51.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 53.1%

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

      1. unpow2 53.1%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 53.1%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 53.1%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 53.1%

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

      1. unpow2 53.1%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 53.1%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 57.3%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 65.4%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 65.4%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 53.1%

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

       1. unpow2 53.1%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 53.1%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 57.3%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 65.4%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 65.9%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 65.9%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 2.2000000000000001e-14 < k

   1. Initial program 43.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l/ 43.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]

      2. associate-*l/ 43.1%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]

      3. associate-*l/ 41.8%

         \[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]

      4. associate-/r/ 41.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]

      5. *-commutative 41.8%

         \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]

      6. associate-/l/ 41.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]

      7. associate-*r* 41.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      8. *-commutative 41.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]

      9. associate-*r* 41.8%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]

      10. *-commutative 41.8%

          \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Simplified 41.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]

   4. Taylor expanded in k around inf 59.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
   5. Step-by-step derivation

      1. unpow2 59.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

      2. *-commutative 59.3%

         \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot \sin k\right)}\right)} \]

   6. Simplified 59.3%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \end{array} \]

Alternative 5: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5e-63)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* k (* (* (/ k l) (/ t l)) (* (sin k) (tan k)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-63) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / (k * (((k / l) * (t / l)) * (sin(k) * tan(k))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 5d-63) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (k * (((k / l) * (t / l)) * (sin(k) * tan(k))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5e-63) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / (k * (((k / l) * (t / l)) * (Math.sin(k) * Math.tan(k))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5e-63:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / (k * (((k / l) * (t / l)) * (math.sin(k) * math.tan(k))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5e-63)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(k * Float64(Float64(Float64(k / l) * Float64(t / l)) * Float64(sin(k) * tan(k)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5e-63)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (k * (((k / l) * (t / l)) * (sin(k) * tan(k))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5e-63], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(k * N[(N[(N[(k / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000002e-63

   1. Initial program 55.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 55.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 56.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 51.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 57.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 65.5%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 52.6%

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

       1. unpow2 52.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 52.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 57.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 65.5%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 66.0%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 5.0000000000000002e-63 < k

   1. Initial program 45.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.2%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 59.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 59.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 59.7%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 67.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 67.9%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 67.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 38.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 22.6%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 22.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 22.2%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 32.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 59.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 59.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 59.7%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 59.7%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 59.7%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 67.9%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 67.9%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 73.1%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 73.1%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]

   11. Taylor expanded in k around 0 59.7%

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

       1. unpow2 59.7%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}\right)} \]

       2. associate-*r* 61.1%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}\right)} \]

       3. unpow2 61.1%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}\right)} \]

       4. times-frac 83.9%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)}\right)} \]

       5. associate-*r/ 78.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\left(k \cdot \frac{t}{\ell}\right)}\right)\right)} \]

       6. *-commutative 78.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(k \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}\right)} \]

       7. associate-*l* 74.3%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]

   13. Simplified 74.3%

       \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]
   14. Step-by-step derivation

       1. expm1-log1p-u 41.6%

          \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)\right)\right)\right)}} \]

       2. expm1-udef 27.3%

          \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)\right)\right)} - 1}} \]

       3. associate-*r* 27.3%

          \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} - 1} \]

       4. *-commutative 27.3%

          \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\left(\tan k \cdot \sin k\right)} \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)\right)} - 1} \]

       5. frac-times 23.9%

          \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \color{blue}{\frac{t \cdot k}{\ell \cdot \ell}}\right)\right)} - 1} \]

       6. *-commutative 23.9%

          \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \frac{\color{blue}{k \cdot t}}{\ell \cdot \ell}\right)\right)} - 1} \]

   15. Applied egg-rr 23.9%

       \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \frac{k \cdot t}{\ell \cdot \ell}\right)\right)} - 1}} \]
   16. Step-by-step derivation

       1. expm1-def 34.4%

          \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \frac{k \cdot t}{\ell \cdot \ell}\right)\right)\right)}} \]

       2. expm1-log1p 62.8%

          \[\leadsto \frac{2}{\color{blue}{\left(\tan k \cdot \sin k\right) \cdot \left(k \cdot \frac{k \cdot t}{\ell \cdot \ell}\right)}} \]

       3. *-commutative 62.8%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \frac{k \cdot t}{\ell \cdot \ell}\right) \cdot \left(\tan k \cdot \sin k\right)}} \]

       4. associate-*l* 62.8%

          \[\leadsto \frac{2}{\color{blue}{k \cdot \left(\frac{k \cdot t}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)}} \]

       5. *-commutative 62.8%

          \[\leadsto \frac{2}{k \cdot \left(\frac{\color{blue}{t \cdot k}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)} \]

       6. times-frac 74.3%

          \[\leadsto \frac{2}{k \cdot \left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\tan k \cdot \sin k\right)\right)} \]

       7. *-commutative 74.3%

          \[\leadsto \frac{2}{k \cdot \left(\left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\left(\sin k \cdot \tan k\right)}\right)} \]

   17. Simplified 74.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(\left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \end{array} \]

Alternative 6: 78.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.3e-63)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (sin k) (* (tan k) (* k (* (/ k l) (/ t l))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e-63) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((k / l) * (t / l)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.3d-63) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (sin(k) * (tan(k) * (k * ((k / l) * (t / l)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.3e-63) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (k * ((k / l) * (t / l)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 3.3e-63:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (k * ((k / l) * (t / l)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.3e-63)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(Float64(k / l) * Float64(t / l))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.3e-63)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((k / l) * (t / l)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.3e-63], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(k / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.29999999999999994e-63

   1. Initial program 55.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 55.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 56.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 51.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 57.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 65.5%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 52.6%

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

       1. unpow2 52.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 52.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 57.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 65.5%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 66.0%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 3.29999999999999994e-63 < k

   1. Initial program 45.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.2%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 59.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 59.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 59.7%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 67.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 67.9%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 67.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 38.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 22.6%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 22.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 22.2%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 32.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 59.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 59.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 59.7%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 59.7%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 59.7%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 67.9%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 67.9%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 73.1%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 73.1%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]

   11. Taylor expanded in k around 0 59.7%

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

       1. unpow2 59.7%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}\right)} \]

       2. associate-*r* 61.1%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}\right)} \]

       3. unpow2 61.1%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}\right)} \]

       4. times-frac 83.9%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)}\right)} \]

       5. associate-*r/ 78.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\left(k \cdot \frac{t}{\ell}\right)}\right)\right)} \]

       6. *-commutative 78.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(k \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}\right)} \]

       7. associate-*l* 74.3%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]

   13. Simplified 74.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{-63}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 7: 79.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{t \cdot \frac{k}{\ell}}{\ell}\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.8e-64)
   (* (/ l k) (/ (/ l k) (pow t 3.0)))
   (/ 2.0 (* (sin k) (* (tan k) (* k (/ (* t (/ k l)) l)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.8e-64) {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	} else {
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.8d-64) then
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    else
        tmp = 2.0d0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.8e-64) {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	} else {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (k * ((t * (k / l)) / l))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6.8e-64:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	else:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (k * ((t * (k / l)) / l))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.8e-64)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(k * Float64(Float64(t * Float64(k / l)) / l)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.8e-64)
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	else
		tmp = 2.0 / (sin(k) * (tan(k) * (k * ((t * (k / l)) / l))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.8e-64], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k * N[(N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.80000000000000024e-64

   1. Initial program 55.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 55.8%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 55.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 56.4%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 55.9%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 51.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 52.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 52.6%

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

      1. unpow2 52.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 52.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 57.0%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 65.5%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 52.6%

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

       1. unpow2 52.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 52.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 57.0%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 65.5%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 66.0%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 66.0%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]

   if 6.80000000000000024e-64 < k

   1. Initial program 45.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 45.2%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 45.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 45.2%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 45.2%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 45.2%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 59.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 59.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 59.7%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 67.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 67.9%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 67.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 38.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 22.6%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 22.2%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 22.2%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 32.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 59.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 59.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 59.7%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 59.7%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 59.7%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 67.9%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 67.9%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 73.1%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 73.1%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]

   11. Taylor expanded in k around 0 59.7%

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

       1. unpow2 59.7%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{{\ell}^{2}}\right)} \]

       2. associate-*r* 61.1%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{k \cdot \left(k \cdot t\right)}}{{\ell}^{2}}\right)} \]

       3. unpow2 61.1%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{k \cdot \left(k \cdot t\right)}{\color{blue}{\ell \cdot \ell}}\right)} \]

       4. times-frac 83.9%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}\right)}\right)} \]

       5. associate-*r/ 78.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\left(k \cdot \frac{t}{\ell}\right)}\right)\right)} \]

       6. *-commutative 78.0%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(k \cdot \frac{t}{\ell}\right) \cdot \frac{k}{\ell}\right)}\right)} \]

       7. associate-*l* 74.3%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]

   13. Simplified 74.3%

       \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k}{\ell}\right)\right)}\right)} \]
   14. Step-by-step derivation

       1. associate-*l/ 75.5%

          \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \color{blue}{\frac{t \cdot \frac{k}{\ell}}{\ell}}\right)\right)} \]

   15. Applied egg-rr 75.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(k \cdot \frac{t \cdot \frac{k}{\ell}}{\ell}\right)\right)}\\ \end{array} \]

Alternative 8: 67.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-86} \lor \neg \left(t \leq 2.65 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.05e-86) (not (<= t 2.65e-73)))
   (* (/ l k) (/ l (* (pow t 3.0) k)))
   (* 2.0 (* (/ l (pow k 4.0)) (/ l t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.05e-86) || !(t <= 2.65e-73)) {
		tmp = (l / k) * (l / (pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * ((l / pow(k, 4.0)) * (l / t));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.05d-86)) .or. (.not. (t <= 2.65d-73))) then
        tmp = (l / k) * (l / ((t ** 3.0d0) * k))
    else
        tmp = 2.0d0 * ((l / (k ** 4.0d0)) * (l / t))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.05e-86) || !(t <= 2.65e-73)) {
		tmp = (l / k) * (l / (Math.pow(t, 3.0) * k));
	} else {
		tmp = 2.0 * ((l / Math.pow(k, 4.0)) * (l / t));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -1.05e-86) or not (t <= 2.65e-73):
		tmp = (l / k) * (l / (math.pow(t, 3.0) * k))
	else:
		tmp = 2.0 * ((l / math.pow(k, 4.0)) * (l / t))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.05e-86) || !(t <= 2.65e-73))
		tmp = Float64(Float64(l / k) * Float64(l / Float64((t ^ 3.0) * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.05e-86) || ~((t <= 2.65e-73)))
		tmp = (l / k) * (l / ((t ^ 3.0) * k));
	else
		tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.05e-86], N[Not[LessEqual[t, 2.65e-73]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e-86 or 2.64999999999999986e-73 < t

   1. Initial program 63.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 63.0%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 63.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 63.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 63.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 63.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 57.9%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 55.2%

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

      1. unpow2 55.2%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 55.2%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 55.2%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 55.2%

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

      1. unpow2 55.2%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 55.2%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 59.9%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 67.5%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   if -1.05e-86 < t < 2.64999999999999986e-73

   1. Initial program 34.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 34.9%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 34.9%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 68.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 68.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 79.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 79.3%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 79.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around 0 56.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 56.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 56.6%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 68.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 68.0%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-86} \lor \neg \left(t \leq 2.65 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 9: 67.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -3.5e-87)
   (* (/ l k) (/ l (* (pow t 3.0) k)))
   (if (<= t 9.6e-73)
     (* 2.0 (* (/ l (pow k 4.0)) (/ l t)))
     (* (/ l k) (/ (/ l k) (pow t 3.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-87) {
		tmp = (l / k) * (l / (pow(t, 3.0) * k));
	} else if (t <= 9.6e-73) {
		tmp = 2.0 * ((l / pow(k, 4.0)) * (l / t));
	} else {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-3.5d-87)) then
        tmp = (l / k) * (l / ((t ** 3.0d0) * k))
    else if (t <= 9.6d-73) then
        tmp = 2.0d0 * ((l / (k ** 4.0d0)) * (l / t))
    else
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -3.5e-87) {
		tmp = (l / k) * (l / (Math.pow(t, 3.0) * k));
	} else if (t <= 9.6e-73) {
		tmp = 2.0 * ((l / Math.pow(k, 4.0)) * (l / t));
	} else {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -3.5e-87:
		tmp = (l / k) * (l / (math.pow(t, 3.0) * k))
	elif t <= 9.6e-73:
		tmp = 2.0 * ((l / math.pow(k, 4.0)) * (l / t))
	else:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -3.5e-87)
		tmp = Float64(Float64(l / k) * Float64(l / Float64((t ^ 3.0) * k)));
	elseif (t <= 9.6e-73)
		tmp = Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t)));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -3.5e-87)
		tmp = (l / k) * (l / ((t ^ 3.0) * k));
	elseif (t <= 9.6e-73)
		tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t));
	else
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -3.5e-87], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.6e-73], N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.50000000000000012e-87

   1. Initial program 65.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 65.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 65.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 65.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 66.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 65.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 62.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 60.7%

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

      1. unpow2 60.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 60.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 60.7%

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

      1. unpow2 60.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 60.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 64.3%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   if -3.50000000000000012e-87 < t < 9.60000000000000022e-73

   1. Initial program 34.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 34.9%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 34.9%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 68.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 68.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 79.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 79.3%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 79.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around 0 56.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   8. Step-by-step derivation

      1. unpow2 56.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

      2. *-commutative 56.6%

         \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

      3. times-frac 68.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   9. Simplified 68.0%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   if 9.60000000000000022e-73 < t

   1. Initial program 60.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 60.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 60.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 60.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 60.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 60.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 53.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 48.6%

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

      1. unpow2 48.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 48.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 48.6%

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

      1. unpow2 48.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 48.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 54.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 67.1%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 48.6%

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

       1. unpow2 48.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 48.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 54.7%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 67.1%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 68.4%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 68.4%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 10: 67.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-86}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -1e-86)
   (* (/ l k) (/ l (* (pow t 3.0) k)))
   (if (<= t 1.4e-74)
     (/ 2.0 (* (/ t l) (/ (pow k 4.0) l)))
     (* (/ l k) (/ (/ l k) (pow t 3.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -1e-86) {
		tmp = (l / k) * (l / (pow(t, 3.0) * k));
	} else if (t <= 1.4e-74) {
		tmp = 2.0 / ((t / l) * (pow(k, 4.0) / l));
	} else {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1d-86)) then
        tmp = (l / k) * (l / ((t ** 3.0d0) * k))
    else if (t <= 1.4d-74) then
        tmp = 2.0d0 / ((t / l) * ((k ** 4.0d0) / l))
    else
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1e-86) {
		tmp = (l / k) * (l / (Math.pow(t, 3.0) * k));
	} else if (t <= 1.4e-74) {
		tmp = 2.0 / ((t / l) * (Math.pow(k, 4.0) / l));
	} else {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -1e-86:
		tmp = (l / k) * (l / (math.pow(t, 3.0) * k))
	elif t <= 1.4e-74:
		tmp = 2.0 / ((t / l) * (math.pow(k, 4.0) / l))
	else:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -1e-86)
		tmp = Float64(Float64(l / k) * Float64(l / Float64((t ^ 3.0) * k)));
	elseif (t <= 1.4e-74)
		tmp = Float64(2.0 / Float64(Float64(t / l) * Float64((k ^ 4.0) / l)));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1e-86)
		tmp = (l / k) * (l / ((t ^ 3.0) * k));
	elseif (t <= 1.4e-74)
		tmp = 2.0 / ((t / l) * ((k ^ 4.0) / l));
	else
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -1e-86], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.4e-74], N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.00000000000000008e-86

   1. Initial program 65.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 65.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 65.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 65.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 66.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 65.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 62.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 60.7%

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

      1. unpow2 60.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 60.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 60.7%

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

      1. unpow2 60.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 60.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 64.3%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   if -1.00000000000000008e-86 < t < 1.39999999999999994e-74

   1. Initial program 34.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 34.9%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 34.9%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 68.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 68.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 79.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 79.3%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 79.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   7. Taylor expanded in k around 0 56.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}} \]
   8. Step-by-step derivation

      1. *-commutative 56.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]

      2. unpow2 56.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{4}}{\color{blue}{\ell \cdot \ell}}} \]

      3. times-frac 68.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}} \]

   9. Simplified 68.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}} \]

   if 1.39999999999999994e-74 < t

   1. Initial program 60.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 60.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 60.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 60.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 60.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 60.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 53.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 48.6%

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

      1. unpow2 48.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 48.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 48.6%

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

      1. unpow2 48.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 48.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 54.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 67.1%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 48.6%

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

       1. unpow2 48.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 48.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 54.7%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 67.1%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 68.4%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 68.4%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-86}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-74}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 11: 67.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-86}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-72}:\\ \;\;\;\;\frac{\ell \cdot 2}{\frac{t}{\frac{\ell}{{k}^{4}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -1.05e-86)
   (* (/ l k) (/ l (* (pow t 3.0) k)))
   (if (<= t 2.7e-72)
     (/ (* l 2.0) (/ t (/ l (pow k 4.0))))
     (* (/ l k) (/ (/ l k) (pow t 3.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.05e-86) {
		tmp = (l / k) * (l / (pow(t, 3.0) * k));
	} else if (t <= 2.7e-72) {
		tmp = (l * 2.0) / (t / (l / pow(k, 4.0)));
	} else {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.05d-86)) then
        tmp = (l / k) * (l / ((t ** 3.0d0) * k))
    else if (t <= 2.7d-72) then
        tmp = (l * 2.0d0) / (t / (l / (k ** 4.0d0)))
    else
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.05e-86) {
		tmp = (l / k) * (l / (Math.pow(t, 3.0) * k));
	} else if (t <= 2.7e-72) {
		tmp = (l * 2.0) / (t / (l / Math.pow(k, 4.0)));
	} else {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -1.05e-86:
		tmp = (l / k) * (l / (math.pow(t, 3.0) * k))
	elif t <= 2.7e-72:
		tmp = (l * 2.0) / (t / (l / math.pow(k, 4.0)))
	else:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -1.05e-86)
		tmp = Float64(Float64(l / k) * Float64(l / Float64((t ^ 3.0) * k)));
	elseif (t <= 2.7e-72)
		tmp = Float64(Float64(l * 2.0) / Float64(t / Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.05e-86)
		tmp = (l / k) * (l / ((t ^ 3.0) * k));
	elseif (t <= 2.7e-72)
		tmp = (l * 2.0) / (t / (l / (k ^ 4.0)));
	else
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -1.05e-86], N[(N[(l / k), $MachinePrecision] * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e-72], N[(N[(l * 2.0), $MachinePrecision] / N[(t / N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.05e-86

   1. Initial program 65.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 65.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 65.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 65.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 66.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 65.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 62.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 60.7%

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

      1. unpow2 60.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 60.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 60.7%

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

      1. unpow2 60.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 60.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 64.3%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   if -1.05e-86 < t < 2.7e-72

   1. Initial program 34.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 34.9%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 34.9%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 68.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 68.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 79.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 79.3%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 79.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 63.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 36.3%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 31.0%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 31.0%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 53.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 68.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 68.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 68.6%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 68.6%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 68.6%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 79.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 79.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 81.6%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 81.6%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]

   11. Taylor expanded in k around 0 56.6%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   12. Step-by-step derivation

       1. unpow2 56.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. times-frac 68.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

       3. *-commutative 68.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

       4. *-commutative 68.0%

          \[\leadsto \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \cdot 2} \]

       5. associate-*l/ 67.4%

          \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \cdot 2 \]

       6. associate-/l* 68.4%

          \[\leadsto \color{blue}{\frac{\ell}{\frac{t}{\frac{\ell}{{k}^{4}}}}} \cdot 2 \]

       7. associate-*l/ 68.4%

          \[\leadsto \color{blue}{\frac{\ell \cdot 2}{\frac{t}{\frac{\ell}{{k}^{4}}}}} \]

   13. Simplified 68.4%

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{\frac{t}{\frac{\ell}{{k}^{4}}}}} \]

   if 2.7e-72 < t

   1. Initial program 60.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 60.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 60.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 60.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 60.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 60.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 53.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 48.6%

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

      1. unpow2 48.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 48.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 48.6%

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

      1. unpow2 48.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 48.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 54.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 67.1%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 48.6%

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

       1. unpow2 48.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 48.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 54.7%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 67.1%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 68.4%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 68.4%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-86}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-72}:\\ \;\;\;\;\frac{\ell \cdot 2}{\frac{t}{\frac{\ell}{{k}^{4}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 12: 66.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3} \cdot k}}{k}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{\ell \cdot 2}{\frac{t}{\frac{\ell}{{k}^{4}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -8e-87)
   (/ (* l (/ l (* (pow t 3.0) k))) k)
   (if (<= t 6.5e-75)
     (/ (* l 2.0) (/ t (/ l (pow k 4.0))))
     (* (/ l k) (/ (/ l k) (pow t 3.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -8e-87) {
		tmp = (l * (l / (pow(t, 3.0) * k))) / k;
	} else if (t <= 6.5e-75) {
		tmp = (l * 2.0) / (t / (l / pow(k, 4.0)));
	} else {
		tmp = (l / k) * ((l / k) / pow(t, 3.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-8d-87)) then
        tmp = (l * (l / ((t ** 3.0d0) * k))) / k
    else if (t <= 6.5d-75) then
        tmp = (l * 2.0d0) / (t / (l / (k ** 4.0d0)))
    else
        tmp = (l / k) * ((l / k) / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -8e-87) {
		tmp = (l * (l / (Math.pow(t, 3.0) * k))) / k;
	} else if (t <= 6.5e-75) {
		tmp = (l * 2.0) / (t / (l / Math.pow(k, 4.0)));
	} else {
		tmp = (l / k) * ((l / k) / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -8e-87:
		tmp = (l * (l / (math.pow(t, 3.0) * k))) / k
	elif t <= 6.5e-75:
		tmp = (l * 2.0) / (t / (l / math.pow(k, 4.0)))
	else:
		tmp = (l / k) * ((l / k) / math.pow(t, 3.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -8e-87)
		tmp = Float64(Float64(l * Float64(l / Float64((t ^ 3.0) * k))) / k);
	elseif (t <= 6.5e-75)
		tmp = Float64(Float64(l * 2.0) / Float64(t / Float64(l / (k ^ 4.0))));
	else
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -8e-87)
		tmp = (l * (l / ((t ^ 3.0) * k))) / k;
	elseif (t <= 6.5e-75)
		tmp = (l * 2.0) / (t / (l / (k ^ 4.0)));
	else
		tmp = (l / k) * ((l / k) / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -8e-87], N[(N[(l * N[(l / N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t, 6.5e-75], N[(N[(l * 2.0), $MachinePrecision] / N[(t / N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.00000000000000014e-87

   1. Initial program 65.4%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 65.4%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 65.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 65.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 66.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 65.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 62.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 60.7%

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

      1. unpow2 60.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 60.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 60.7%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 60.7%

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

      1. unpow2 60.7%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 60.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 64.3%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 67.9%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]
   10. Step-by-step derivation

       1. associate-*l/ 68.0%

          \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{k}} \]

   11. Applied egg-rr 68.0%

       \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot {t}^{3}}}{k}} \]

   if -8.00000000000000014e-87 < t < 6.5000000000000002e-75

   1. Initial program 34.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. *-commutative 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

      2. associate-*l* 34.9%

         \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

      3. associate-*r* 34.9%

         \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

      4. +-commutative 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      5. associate-+r+ 34.9%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

      6. metadata-eval 34.9%

         \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. Taylor expanded in k around inf 68.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
   5. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. unpow2 68.6%

         \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

      3. times-frac 79.3%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

      4. unpow2 79.3%

         \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. Simplified 79.3%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 63.6%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-udef 36.3%

         \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]

      3. frac-times 31.0%

         \[\leadsto \frac{2}{e^{\mathsf{log1p}\left(\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1} \]

   8. Applied egg-rr 31.0%

      \[\leadsto \frac{2}{\color{blue}{e^{\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} - 1}} \]
   9. Step-by-step derivation

      1. expm1-def 53.0%

         \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)\right)}} \]

      2. expm1-log1p 68.6%

         \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]

      3. *-commutative 68.6%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}}} \]

      4. associate-*l* 68.6%

         \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \left(k \cdot k\right)}{\ell \cdot \ell}\right)}} \]

      5. unpow2 68.6%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{t \cdot \color{blue}{{k}^{2}}}{\ell \cdot \ell}\right)} \]

      6. *-commutative 68.6%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \frac{\color{blue}{{k}^{2} \cdot t}}{\ell \cdot \ell}\right)} \]

      7. times-frac 79.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\right)} \]

      8. unpow2 79.3%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot \frac{t}{\ell}\right)\right)} \]

      9. associate-*l/ 81.6%

         \[\leadsto \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{t}{\ell}\right)\right)} \]

   10. Simplified 81.6%

       \[\leadsto \frac{2}{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{t}{\ell}\right)\right)}} \]

   11. Taylor expanded in k around 0 56.6%

       \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
   12. Step-by-step derivation

       1. unpow2 56.6%

          \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

       2. times-frac 68.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right)} \]

       3. *-commutative 68.0%

          \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

       4. *-commutative 68.0%

          \[\leadsto \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \cdot 2} \]

       5. associate-*l/ 67.4%

          \[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}} \cdot 2 \]

       6. associate-/l* 68.4%

          \[\leadsto \color{blue}{\frac{\ell}{\frac{t}{\frac{\ell}{{k}^{4}}}}} \cdot 2 \]

       7. associate-*l/ 68.4%

          \[\leadsto \color{blue}{\frac{\ell \cdot 2}{\frac{t}{\frac{\ell}{{k}^{4}}}}} \]

   13. Simplified 68.4%

       \[\leadsto \color{blue}{\frac{\ell \cdot 2}{\frac{t}{\frac{\ell}{{k}^{4}}}}} \]

   if 6.5000000000000002e-75 < t

   1. Initial program 60.2%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*l* 60.2%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

      2. associate-/l/ 60.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]

      3. *-commutative 60.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]

      4. associate-*r/ 60.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]

      5. associate-/l* 60.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]

      6. associate-/r/ 53.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \]

   4. Taylor expanded in k around 0 48.6%

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

      1. unpow2 48.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 48.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

   6. Simplified 48.6%

      \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot {t}^{3}}} \]

   7. Taylor expanded in l around 0 48.6%

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

      1. unpow2 48.6%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. unpow2 48.6%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

      3. associate-*r* 54.7%

         \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

      4. times-frac 67.1%

         \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   9. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

   10. Taylor expanded in l around 0 48.6%

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

       1. unpow2 48.6%

          \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

       2. unpow2 48.6%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}} \]

       3. associate-*r* 54.7%

          \[\leadsto \frac{\ell \cdot \ell}{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}} \]

       4. times-frac 67.1%

          \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}} \]

       5. associate-/r* 68.4%

          \[\leadsto \frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \]

   12. Simplified 68.4%

       \[\leadsto \color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-87}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3} \cdot k}}{k}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{\ell \cdot 2}{\frac{t}{\frac{\ell}{{k}^{4}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 13: 55.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l (pow k 4.0)) (/ l t))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l / pow(k, 4.0)) * (l / t));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l / (k ** 4.0d0)) * (l / t))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l / Math.pow(k, 4.0)) * (l / t));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((l / math.pow(k, 4.0)) * (l / t))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t)))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 52.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*l* 49.5%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   3. associate-*r* 49.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. +-commutative 49.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. associate-+r+ 49.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. metadata-eval 49.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

3. Simplified 49.5%

   \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

4. Taylor expanded in k around inf 60.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
5. Step-by-step derivation

   1. *-commutative 60.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. unpow2 60.4%

      \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

   3. times-frac 66.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   4. unpow2 66.9%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

6. Simplified 66.9%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in k around 0 52.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 52.8%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 52.8%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

   3. times-frac 57.9%

      \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

9. Simplified 57.9%

   \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

10. Final simplification 57.9%

    \[\leadsto 2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right) \]

Alternative 14: 51.8% accurate, 28.1× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ (* (* l l) 2.0) (* (* k k) (* k (* t k)))))

C

k = abs(k);
double code(double t, double l, double k) {
	return ((l * l) * 2.0) / ((k * k) * (k * (t * k)));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((l * l) * 2.0d0) / ((k * k) * (k * (t * k)))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return ((l * l) * 2.0) / ((k * k) * (k * (t * k)));
}

Python

k = abs(k)
def code(t, l, k):
	return ((l * l) * 2.0) / ((k * k) * (k * (t * k)))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(Float64(Float64(l * l) * 2.0) / Float64(Float64(k * k) * Float64(k * Float64(t * k))))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = ((l * l) * 2.0) / ((k * k) * (k * (t * k)));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(l * l), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.6%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
2. Step-by-step derivation

   1. *-commutative 52.6%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]

   2. associate-*l* 49.5%

      \[\leadsto \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]

   3. associate-*r* 49.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

   4. +-commutative 49.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   5. associate-+r+ 49.5%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   6. metadata-eval 49.5%

      \[\leadsto \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

3. Simplified 49.5%

   \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]

4. Taylor expanded in k around inf 60.4%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}}} \cdot \left(\sin k \cdot \tan k\right)} \]
5. Step-by-step derivation

   1. *-commutative 60.4%

      \[\leadsto \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   2. unpow2 60.4%

      \[\leadsto \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]

   3. times-frac 66.9%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

   4. unpow2 66.9%

      \[\leadsto \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

6. Simplified 66.9%

   \[\leadsto \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]

7. Taylor expanded in t around 0 59.0%

   \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
8. Step-by-step derivation

   1. associate-*r/ 59.0%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]

   2. *-commutative 59.0%

      \[\leadsto \frac{2 \cdot \color{blue}{\left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

   3. unpow2 59.0%

      \[\leadsto \frac{2 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]

   4. *-commutative 59.0%

      \[\leadsto \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]

   5. associate-*r* 59.0%

      \[\leadsto \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   6. unpow2 59.0%

      \[\leadsto \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {\sin k}^{2}} \]

   7. associate-*l* 61.3%

      \[\leadsto \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot {\sin k}^{2}} \]

9. Simplified 61.3%

   \[\leadsto \color{blue}{\frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}} \]

10. Taylor expanded in k around 0 54.5%

    \[\leadsto \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \color{blue}{{k}^{2}}} \]
11. Step-by-step derivation

    1. unpow2 54.5%

       \[\leadsto \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

12. Simplified 54.5%

    \[\leadsto \frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \cos k\right)}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \color{blue}{\left(k \cdot k\right)}} \]

13. Taylor expanded in k around 0 53.3%

    \[\leadsto \frac{2 \cdot \color{blue}{{\ell}^{2}}}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)} \]
14. Step-by-step derivation

    1. unpow2 53.3%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)} \]

15. Simplified 53.3%

    \[\leadsto \frac{2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)} \]

16. Final simplification 53.3%

    \[\leadsto \frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)} \]

Reproduce

herbie shell --seed 2023217 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))