Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 99.1%

Time: 23.1s

Alternatives: 13

Speedup: 3.8×

• 35.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 34.5% 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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \sqrt{t_m}\right)\right)}^{2}}{\cos k}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (/ (pow (* (/ k l) (* (sin k) (sqrt t_m))) 2.0) (cos k)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (pow(((k / l) * (sin(k) * sqrt(t_m))), 2.0) / cos(k)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / ((((k / l) * (sin(k) * sqrt(t_m))) ** 2.0d0) / cos(k)))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (Math.pow(((k / l) * (Math.sin(k) * Math.sqrt(t_m))), 2.0) / Math.cos(k)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (math.pow(((k / l) * (math.sin(k) * math.sqrt(t_m))), 2.0) / math.cos(k)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64((Float64(Float64(k / l) * Float64(sin(k) * sqrt(t_m))) ^ 2.0) / cos(k))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / ((((k / l) * (sin(k) * sqrt(t_m))) ^ 2.0) / cos(k)));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[Power[N[(N[(k / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in t around 0 74.9%

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

   1. times-frac 77.5%

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

4. Simplified 77.5%

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

   1. add-sqr-sqrt 77.5%

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

   2. sqrt-div 77.5%

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

   3. unpow2 77.5%

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

   4. sqrt-prod 41.1%

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

   5. add-sqr-sqrt 56.4%

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

   6. pow2 56.4%

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

   7. sqrt-prod 28.7%

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

   8. add-sqr-sqrt 51.4%

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

   9. sqrt-div 51.4%

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

   10. unpow2 51.4%

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

   11. sqrt-prod 24.8%

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

   12. add-sqr-sqrt 50.3%

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

   13. pow2 50.3%

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

   14. sqrt-prod 43.5%

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

   15. add-sqr-sqrt 92.8%

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

6. Applied egg-rr 92.8%

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

   1. expm1-log1p-u 74.2%

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

   2. expm1-udef 34.5%

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

8. Applied egg-rr 26.4%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 40.8%

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

10. Simplified 40.8%

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

11. Final simplification 40.8%

    \[\leadsto \frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}^{2}}{\cos k}} \]

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \frac{2}{\frac{{\left(\sqrt{t_m} \cdot \left(\frac{k}{\ell} \cdot \sin k\right)\right)}^{2}}{\cos k}} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ 2.0 (/ (pow (* (sqrt t_m) (* (/ k l) (sin k))) 2.0) (cos k)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (pow((sqrt(t_m) * ((k / l) * sin(k))), 2.0) / cos(k)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 / (((sqrt(t_m) * ((k / l) * sin(k))) ** 2.0d0) / cos(k)))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 / (Math.pow((Math.sqrt(t_m) * ((k / l) * Math.sin(k))), 2.0) / Math.cos(k)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 / (math.pow((math.sqrt(t_m) * ((k / l) * math.sin(k))), 2.0) / math.cos(k)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 / Float64((Float64(sqrt(t_m) * Float64(Float64(k / l) * sin(k))) ^ 2.0) / cos(k))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 / (((sqrt(t_m) * ((k / l) * sin(k))) ^ 2.0) / cos(k)));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 / N[(N[Power[N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

2. Taylor expanded in t around 0 74.9%

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

   1. times-frac 77.5%

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

4. Simplified 77.5%

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

   1. add-sqr-sqrt 77.5%

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

   2. sqrt-div 77.5%

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

   3. unpow2 77.5%

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

   4. sqrt-prod 41.1%

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

   5. add-sqr-sqrt 56.4%

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

   6. pow2 56.4%

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

   7. sqrt-prod 28.7%

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

   8. add-sqr-sqrt 51.4%

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

   9. sqrt-div 51.4%

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

   10. unpow2 51.4%

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

   11. sqrt-prod 24.8%

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

   12. add-sqr-sqrt 50.3%

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

   13. pow2 50.3%

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

   14. sqrt-prod 43.5%

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

   15. add-sqr-sqrt 92.8%

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

6. Applied egg-rr 92.8%

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

   1. expm1-log1p-u 74.2%

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

   2. expm1-udef 34.5%

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

8. Applied egg-rr 26.4%

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

   1. expm1-def 34.5%

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

   2. expm1-log1p 40.8%

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

10. Simplified 40.8%

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

11. Taylor expanded in k around inf 40.0%

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

    1. associate-*l/ 40.8%

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

    2. *-commutative 40.8%

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

    3. *-commutative 40.8%

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

13. Simplified 40.8%

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

14. Final simplification 40.8%

    \[\leadsto \frac{2}{\frac{{\left(\sqrt{t} \cdot \left(\frac{k}{\ell} \cdot \sin k\right)\right)}^{2}}{\cos k}} \]

Alternative 3: 85.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t_m}\right)\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t_m \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{t_m \cdot {\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.2e-62)
    (/ 2.0 (/ (pow (* (/ k l) (* k (sqrt t_m))) 2.0) (cos k)))
    (if (<= k 1.1e+95)
      (/ 2.0 (/ (* (* k (/ k l)) (* t_m (pow (sin k) 2.0))) (* l (cos k))))
      (* (cos k) (/ 2.0 (* t_m (pow (* (/ k l) (sin k)) 2.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.2e-62) {
		tmp = 2.0 / (pow(((k / l) * (k * sqrt(t_m))), 2.0) / cos(k));
	} else if (k <= 1.1e+95) {
		tmp = 2.0 / (((k * (k / l)) * (t_m * pow(sin(k), 2.0))) / (l * cos(k)));
	} else {
		tmp = cos(k) * (2.0 / (t_m * pow(((k / l) * sin(k)), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.2d-62) then
        tmp = 2.0d0 / ((((k / l) * (k * sqrt(t_m))) ** 2.0d0) / cos(k))
    else if (k <= 1.1d+95) then
        tmp = 2.0d0 / (((k * (k / l)) * (t_m * (sin(k) ** 2.0d0))) / (l * cos(k)))
    else
        tmp = cos(k) * (2.0d0 / (t_m * (((k / l) * sin(k)) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.2e-62) {
		tmp = 2.0 / (Math.pow(((k / l) * (k * Math.sqrt(t_m))), 2.0) / Math.cos(k));
	} else if (k <= 1.1e+95) {
		tmp = 2.0 / (((k * (k / l)) * (t_m * Math.pow(Math.sin(k), 2.0))) / (l * Math.cos(k)));
	} else {
		tmp = Math.cos(k) * (2.0 / (t_m * Math.pow(((k / l) * Math.sin(k)), 2.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.2e-62:
		tmp = 2.0 / (math.pow(((k / l) * (k * math.sqrt(t_m))), 2.0) / math.cos(k))
	elif k <= 1.1e+95:
		tmp = 2.0 / (((k * (k / l)) * (t_m * math.pow(math.sin(k), 2.0))) / (l * math.cos(k)))
	else:
		tmp = math.cos(k) * (2.0 / (t_m * math.pow(((k / l) * math.sin(k)), 2.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.2e-62)
		tmp = Float64(2.0 / Float64((Float64(Float64(k / l) * Float64(k * sqrt(t_m))) ^ 2.0) / cos(k)));
	elseif (k <= 1.1e+95)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(k / l)) * Float64(t_m * (sin(k) ^ 2.0))) / Float64(l * cos(k))));
	else
		tmp = Float64(cos(k) * Float64(2.0 / Float64(t_m * (Float64(Float64(k / l) * sin(k)) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.2e-62)
		tmp = 2.0 / ((((k / l) * (k * sqrt(t_m))) ^ 2.0) / cos(k));
	elseif (k <= 1.1e+95)
		tmp = 2.0 / (((k * (k / l)) * (t_m * (sin(k) ^ 2.0))) / (l * cos(k)));
	else
		tmp = cos(k) * (2.0 / (t_m * (((k / l) * sin(k)) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.2e-62], N[(2.0 / N[(N[Power[N[(N[(k / l), $MachinePrecision] * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.1e+95], N[(2.0 / N[(N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[k], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[N[(N[(k / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.19999999999999992e-62

   1. Initial program 38.3%

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

   2. Taylor expanded in t around 0 74.9%

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

      1. times-frac 76.6%

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

   4. Simplified 76.6%

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

      1. add-sqr-sqrt 76.5%

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

      2. sqrt-div 76.5%

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

      3. unpow2 76.5%

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

      4. sqrt-prod 24.2%

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

      5. add-sqr-sqrt 46.2%

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

      6. pow2 46.2%

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

      7. sqrt-prod 25.1%

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

      8. add-sqr-sqrt 48.4%

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

      9. sqrt-div 48.4%

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

      10. unpow2 48.4%

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

      11. sqrt-prod 10.1%

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

      12. add-sqr-sqrt 46.8%

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

      13. pow2 46.8%

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

      14. sqrt-prod 44.1%

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

      15. add-sqr-sqrt 90.9%

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

   6. Applied egg-rr 90.9%

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

      1. expm1-log1p-u 76.3%

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

      2. expm1-udef 31.5%

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

   8. Applied egg-rr 25.5%

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

      1. expm1-def 30.8%

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

      2. expm1-log1p 35.8%

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

   10. Simplified 35.8%

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

   11. Taylor expanded in k around 0 28.5%

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

   if 1.19999999999999992e-62 < k < 1.0999999999999999e95

   1. Initial program 28.2%

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

   2. Taylor expanded in t around 0 87.3%

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

      1. times-frac 87.6%

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

   4. Simplified 87.6%

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

      1. add-sqr-sqrt 87.6%

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

      2. sqrt-div 87.6%

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

      3. unpow2 87.6%

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

      4. sqrt-prod 87.5%

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

      5. add-sqr-sqrt 87.6%

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

      6. pow2 87.6%

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

      7. sqrt-prod 46.8%

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

      8. add-sqr-sqrt 56.7%

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

      9. sqrt-div 56.7%

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

      10. unpow2 56.7%

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

      11. sqrt-prod 56.7%

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

      12. add-sqr-sqrt 56.7%

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

      13. pow2 56.7%

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

      14. sqrt-prod 52.7%

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

      15. add-sqr-sqrt 96.5%

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

   6. Applied egg-rr 96.5%

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

      1. associate-*r/ 96.6%

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

      2. frac-times 99.6%

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

   8. Applied egg-rr 99.6%

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

   if 1.0999999999999999e95 < k

   1. Initial program 37.1%

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

   2. Taylor expanded in t around 0 66.2%

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

      1. times-frac 74.0%

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

   4. Simplified 74.0%

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

      1. add-sqr-sqrt 74.1%

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

      2. sqrt-div 74.1%

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

      3. unpow2 74.1%

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

      4. sqrt-prod 74.0%

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

      5. add-sqr-sqrt 74.1%

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

      6. pow2 74.1%

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

      7. sqrt-prod 30.3%

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

      8. add-sqr-sqrt 59.4%

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

      9. sqrt-div 59.4%

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

      10. unpow2 59.4%

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

      11. sqrt-prod 59.5%

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

      12. add-sqr-sqrt 59.6%

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

      13. pow2 59.6%

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

      14. sqrt-prod 34.7%

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

      15. add-sqr-sqrt 97.5%

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

   6. Applied egg-rr 97.5%

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

      1. expm1-log1p-u 68.0%

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

      2. expm1-udef 51.1%

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

   8. Applied egg-rr 31.9%

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

      1. expm1-def 42.2%

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

      2. expm1-log1p 51.8%

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

   10. Simplified 51.8%

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

       1. associate-/r/ 51.8%

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

       2. associate-*r* 51.9%

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

       3. unpow-prod-down 49.8%

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

       4. pow2 49.8%

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

       5. add-sqr-sqrt 97.7%

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

   12. Applied egg-rr 97.7%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}{\cos k}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{t \cdot {\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}\\ \end{array} \]

Alternative 4: 84.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 225:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t_m}\right)\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_m \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 225.0)
    (/ 2.0 (/ (pow (* (/ k l) (* k (sqrt t_m))) 2.0) (cos k)))
    (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t_m (pow (sin k) 2.0))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 225.0) {
		tmp = 2.0 / (pow(((k / l) * (k * sqrt(t_m))), 2.0) / cos(k));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t_m * pow(sin(k), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 225.0d0) then
        tmp = 2.0d0 / ((((k / l) * (k * sqrt(t_m))) ** 2.0d0) / cos(k))
    else
        tmp = 2.0d0 * (((l / k) * (l / k)) * (cos(k) / (t_m * (sin(k) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 225.0) {
		tmp = 2.0 / (Math.pow(((k / l) * (k * Math.sqrt(t_m))), 2.0) / Math.cos(k));
	} else {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t_m * Math.pow(Math.sin(k), 2.0))));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 225.0:
		tmp = 2.0 / (math.pow(((k / l) * (k * math.sqrt(t_m))), 2.0) / math.cos(k))
	else:
		tmp = 2.0 * (((l / k) * (l / k)) * (math.cos(k) / (t_m * math.pow(math.sin(k), 2.0))))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 225.0)
		tmp = Float64(2.0 / Float64((Float64(Float64(k / l) * Float64(k * sqrt(t_m))) ^ 2.0) / cos(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t_m * (sin(k) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 225.0)
		tmp = 2.0 / ((((k / l) * (k * sqrt(t_m))) ^ 2.0) / cos(k));
	else
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t_m * (sin(k) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 225.0], N[(2.0 / N[(N[Power[N[(N[(k / l), $MachinePrecision] * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t$95$m * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 225

   1. Initial program 38.3%

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

   2. Taylor expanded in t around 0 76.1%

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

      1. times-frac 77.6%

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

   4. Simplified 77.6%

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

      1. add-sqr-sqrt 77.6%

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

      2. sqrt-div 77.6%

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

      3. unpow2 77.6%

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

      4. sqrt-prod 28.8%

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

      5. add-sqr-sqrt 49.3%

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

      6. pow2 49.3%

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

      7. sqrt-prod 27.0%

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

      8. add-sqr-sqrt 48.8%

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

      9. sqrt-div 48.8%

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

      10. unpow2 48.8%

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

      11. sqrt-prod 13.1%

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

      12. add-sqr-sqrt 47.3%

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

      13. pow2 47.3%

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

      14. sqrt-prod 44.8%

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

      15. add-sqr-sqrt 91.5%

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

   6. Applied egg-rr 91.5%

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

      1. expm1-log1p-u 75.8%

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

      2. expm1-udef 31.5%

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

   8. Applied egg-rr 25.9%

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

      1. expm1-def 31.3%

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

      2. expm1-log1p 36.0%

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

   10. Simplified 36.0%

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

   11. Taylor expanded in k around 0 29.2%

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

   if 225 < k

   1. Initial program 32.5%

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

      1. associate-/r* 32.5%

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

      2. *-commutative 32.5%

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

      3. associate-*l* 32.4%

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

      4. associate-*l/ 32.5%

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

      5. +-commutative 32.5%

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

      6. unpow2 32.5%

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

      7. sqr-neg 32.5%

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

      8. distribute-frac-neg 32.5%

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

      9. distribute-frac-neg 32.5%

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

      10. unpow2 32.5%

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

      11. associate--l+ 43.4%

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

      12. metadata-eval 43.4%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 43.4%

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

      14. unpow2 43.4%

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

      15. distribute-frac-neg 43.4%

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

      16. distribute-frac-neg 43.4%

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

   3. Simplified 43.4%

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

   4. Taylor expanded in k around inf 71.4%

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

      1. times-frac 77.1%

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

   6. Simplified 77.1%

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

      1. add-sqr-sqrt 77.1%

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

      2. sqrt-div 77.1%

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

      3. pow2 77.1%

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

      4. sqrt-prod 33.8%

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

      5. add-sqr-sqrt 59.1%

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

      6. unpow2 59.1%

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

      7. sqrt-prod 59.1%

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

      8. add-sqr-sqrt 59.1%

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

      9. sqrt-div 59.1%

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

      10. pow2 59.1%

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

      11. sqrt-prod 36.8%

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

      12. add-sqr-sqrt 83.4%

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

      13. unpow2 83.4%

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

      14. sqrt-prod 96.5%

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

      15. add-sqr-sqrt 96.6%

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

   8. Applied egg-rr 96.6%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 225:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 5: 84.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{{t_m}^{-0.5}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{t_m \cdot {\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.2e-8)
    (* 2.0 (pow (* (/ l k) (/ (pow t_m -0.5) k)) 2.0))
    (* (cos k) (/ 2.0 (* t_m (pow (* (/ k l) (sin k)) 2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-8) {
		tmp = 2.0 * pow(((l / k) * (pow(t_m, -0.5) / k)), 2.0);
	} else {
		tmp = cos(k) * (2.0 / (t_m * pow(((k / l) * sin(k)), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d-8) then
        tmp = 2.0d0 * (((l / k) * ((t_m ** (-0.5d0)) / k)) ** 2.0d0)
    else
        tmp = cos(k) * (2.0d0 / (t_m * (((k / l) * sin(k)) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-8) {
		tmp = 2.0 * Math.pow(((l / k) * (Math.pow(t_m, -0.5) / k)), 2.0);
	} else {
		tmp = Math.cos(k) * (2.0 / (t_m * Math.pow(((k / l) * Math.sin(k)), 2.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.2e-8:
		tmp = 2.0 * math.pow(((l / k) * (math.pow(t_m, -0.5) / k)), 2.0)
	else:
		tmp = math.cos(k) * (2.0 / (t_m * math.pow(((k / l) * math.sin(k)), 2.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.2e-8)
		tmp = Float64(2.0 * (Float64(Float64(l / k) * Float64((t_m ^ -0.5) / k)) ^ 2.0));
	else
		tmp = Float64(cos(k) * Float64(2.0 / Float64(t_m * (Float64(Float64(k / l) * sin(k)) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.2e-8)
		tmp = 2.0 * (((l / k) * ((t_m ^ -0.5) / k)) ^ 2.0);
	else
		tmp = cos(k) * (2.0 / (t_m * (((k / l) * sin(k)) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.2e-8], N[(2.0 * N[Power[N[(N[(l / k), $MachinePrecision] * N[(N[Power[t$95$m, -0.5], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[k], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[N[(N[(k / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.1999999999999998e-8

   1. Initial program 37.6%

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

      1. associate-/r* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-*l* 36.9%

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

      4. associate-*l/ 38.0%

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

      5. +-commutative 38.0%

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

      6. unpow2 38.0%

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

      7. sqr-neg 38.0%

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

      8. distribute-frac-neg 38.0%

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

      9. distribute-frac-neg 38.0%

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

      10. unpow2 38.0%

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

      11. associate--l+ 43.6%

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

      12. metadata-eval 43.6%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 43.6%

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

      14. unpow2 43.6%

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

      15. distribute-frac-neg 43.6%

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

      16. distribute-frac-neg 43.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in k around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-/r* 62.9%

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

   6. Simplified 62.9%

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

      1. add-sqr-sqrt 31.8%

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

      2. sqrt-div 25.4%

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

      3. sqrt-div 19.4%

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

      4. pow2 19.4%

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

      5. sqrt-prod 9.7%

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

      6. add-sqr-sqrt 13.0%

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

      7. sqrt-pow1 13.0%

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

      8. metadata-eval 13.0%

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

      9. sqrt-div 13.0%

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

      10. sqrt-div 13.1%

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

      11. pow2 13.1%

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

      12. sqrt-prod 10.8%

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

      13. add-sqr-sqrt 24.2%

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

      14. sqrt-pow1 26.8%

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

      15. metadata-eval 26.8%

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

   8. Applied egg-rr 26.8%

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

      1. unpow2 26.8%

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

      2. associate-/l/ 26.9%

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

   10. Simplified 26.9%

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

   11. Taylor expanded in l around 0 27.8%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. associate-*l/ 26.8%

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

   13. Simplified 26.8%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]
   14. Step-by-step derivation

       1. *-commutative 26.8%

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

       2. unpow2 26.8%

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

       3. times-frac 28.8%

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

       4. inv-pow 28.8%

          \[\leadsto 2 \cdot {\left(\frac{\sqrt{\color{blue}{{t}^{-1}}}}{k} \cdot \frac{\ell}{k}\right)}^{2} \]

       5. sqrt-pow1 28.8%

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

       6. metadata-eval 28.8%

          \[\leadsto 2 \cdot {\left(\frac{{t}^{\color{blue}{-0.5}}}{k} \cdot \frac{\ell}{k}\right)}^{2} \]

   15. Applied egg-rr 28.8%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{{t}^{-0.5}}{k} \cdot \frac{\ell}{k}\right)}}^{2} \]

   if 2.1999999999999998e-8 < k

   1. Initial program 34.5%

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

   2. Taylor expanded in t around 0 72.3%

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

      1. times-frac 77.7%

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

   4. Simplified 77.7%

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

      1. add-sqr-sqrt 77.8%

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

      2. sqrt-div 77.8%

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

      3. unpow2 77.8%

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

      4. sqrt-prod 77.7%

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

      5. add-sqr-sqrt 77.8%

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

      6. pow2 77.8%

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

      7. sqrt-prod 34.3%

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

      8. add-sqr-sqrt 58.8%

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

      9. sqrt-div 58.8%

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

      10. unpow2 58.8%

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

      11. sqrt-prod 59.0%

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

      12. add-sqr-sqrt 59.0%

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

      13. pow2 59.0%

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

      14. sqrt-prod 40.1%

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

      15. add-sqr-sqrt 96.7%

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

   6. Applied egg-rr 96.7%

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

      1. expm1-log1p-u 70.4%

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

      2. expm1-udef 43.6%

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

   8. Applied egg-rr 28.8%

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

      1. expm1-def 43.9%

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

      2. expm1-log1p 54.9%

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

   10. Simplified 54.9%

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

       1. associate-/r/ 54.9%

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

       2. associate-*r* 54.9%

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

       3. unpow-prod-down 52.1%

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

       4. pow2 52.1%

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

       5. add-sqr-sqrt 96.9%

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

   12. Applied egg-rr 96.9%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{{t}^{-0.5}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{t \cdot {\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}\\ \end{array} \]

Alternative 6: 84.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 225:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t_m}\right)\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{t_m \cdot {\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 225.0)
    (/ 2.0 (/ (pow (* (/ k l) (* k (sqrt t_m))) 2.0) (cos k)))
    (* (cos k) (/ 2.0 (* t_m (pow (* (/ k l) (sin k)) 2.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 225.0) {
		tmp = 2.0 / (pow(((k / l) * (k * sqrt(t_m))), 2.0) / cos(k));
	} else {
		tmp = cos(k) * (2.0 / (t_m * pow(((k / l) * sin(k)), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 225.0d0) then
        tmp = 2.0d0 / ((((k / l) * (k * sqrt(t_m))) ** 2.0d0) / cos(k))
    else
        tmp = cos(k) * (2.0d0 / (t_m * (((k / l) * sin(k)) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 225.0) {
		tmp = 2.0 / (Math.pow(((k / l) * (k * Math.sqrt(t_m))), 2.0) / Math.cos(k));
	} else {
		tmp = Math.cos(k) * (2.0 / (t_m * Math.pow(((k / l) * Math.sin(k)), 2.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 225.0:
		tmp = 2.0 / (math.pow(((k / l) * (k * math.sqrt(t_m))), 2.0) / math.cos(k))
	else:
		tmp = math.cos(k) * (2.0 / (t_m * math.pow(((k / l) * math.sin(k)), 2.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 225.0)
		tmp = Float64(2.0 / Float64((Float64(Float64(k / l) * Float64(k * sqrt(t_m))) ^ 2.0) / cos(k)));
	else
		tmp = Float64(cos(k) * Float64(2.0 / Float64(t_m * (Float64(Float64(k / l) * sin(k)) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 225.0)
		tmp = 2.0 / ((((k / l) * (k * sqrt(t_m))) ^ 2.0) / cos(k));
	else
		tmp = cos(k) * (2.0 / (t_m * (((k / l) * sin(k)) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 225.0], N[(2.0 / N[(N[Power[N[(N[(k / l), $MachinePrecision] * N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[k], $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[N[(N[(k / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 225

   1. Initial program 38.3%

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

   2. Taylor expanded in t around 0 76.1%

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

      1. times-frac 77.6%

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

   4. Simplified 77.6%

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

      1. add-sqr-sqrt 77.6%

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

      2. sqrt-div 77.6%

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

      3. unpow2 77.6%

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

      4. sqrt-prod 28.8%

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

      5. add-sqr-sqrt 49.3%

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

      6. pow2 49.3%

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

      7. sqrt-prod 27.0%

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

      8. add-sqr-sqrt 48.8%

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

      9. sqrt-div 48.8%

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

      10. unpow2 48.8%

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

      11. sqrt-prod 13.1%

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

      12. add-sqr-sqrt 47.3%

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

      13. pow2 47.3%

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

      14. sqrt-prod 44.8%

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

      15. add-sqr-sqrt 91.5%

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

   6. Applied egg-rr 91.5%

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

      1. expm1-log1p-u 75.8%

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

      2. expm1-udef 31.5%

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

   8. Applied egg-rr 25.9%

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

      1. expm1-def 31.3%

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

      2. expm1-log1p 36.0%

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

   10. Simplified 36.0%

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

   11. Taylor expanded in k around 0 29.2%

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

   if 225 < k

   1. Initial program 32.5%

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

   2. Taylor expanded in t around 0 71.4%

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

      1. times-frac 77.1%

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

   4. Simplified 77.1%

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

      1. add-sqr-sqrt 77.1%

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

      2. sqrt-div 77.1%

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

      3. unpow2 77.1%

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

      4. sqrt-prod 77.1%

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

      5. add-sqr-sqrt 77.1%

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

      6. pow2 77.1%

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

      7. sqrt-prod 33.8%

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

      8. add-sqr-sqrt 59.1%

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

      9. sqrt-div 59.1%

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

      10. unpow2 59.1%

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

      11. sqrt-prod 59.2%

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

      12. add-sqr-sqrt 59.2%

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

      13. pow2 59.2%

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

      14. sqrt-prod 39.8%

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

      15. add-sqr-sqrt 96.6%

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

   6. Applied egg-rr 96.6%

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

      1. expm1-log1p-u 69.5%

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

      2. expm1-udef 43.4%

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

   8. Applied egg-rr 28.2%

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

      1. expm1-def 43.7%

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

      2. expm1-log1p 55.0%

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

   10. Simplified 55.0%

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

       1. associate-/r/ 55.1%

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

       2. associate-*r* 55.1%

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

       3. unpow-prod-down 52.1%

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

       4. pow2 52.1%

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

       5. add-sqr-sqrt 96.8%

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

   12. Applied egg-rr 96.8%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 225:\\ \;\;\;\;\frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \left(k \cdot \sqrt{t}\right)\right)}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\cos k \cdot \frac{2}{t \cdot {\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}\\ \end{array} \]

Alternative 7: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{{t_m}^{-0.5}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t_m \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 2.2e-8)
    (* 2.0 (pow (* (/ l k) (/ (pow t_m -0.5) k)) 2.0))
    (/
     2.0
     (*
      (* (/ k l) (/ k l))
      (/ (* t_m (- 0.5 (/ (cos (* 2.0 k)) 2.0))) (cos k)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-8) {
		tmp = 2.0 * pow(((l / k) * (pow(t_m, -0.5) / k)), 2.0);
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_m * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 2.2d-8) then
        tmp = 2.0d0 * (((l / k) * ((t_m ** (-0.5d0)) / k)) ** 2.0d0)
    else
        tmp = 2.0d0 / (((k / l) * (k / l)) * ((t_m * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 2.2e-8) {
		tmp = 2.0 * Math.pow(((l / k) * (Math.pow(t_m, -0.5) / k)), 2.0);
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_m * (0.5 - (Math.cos((2.0 * k)) / 2.0))) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 2.2e-8:
		tmp = 2.0 * math.pow(((l / k) * (math.pow(t_m, -0.5) / k)), 2.0)
	else:
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_m * (0.5 - (math.cos((2.0 * k)) / 2.0))) / math.cos(k)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 2.2e-8)
		tmp = Float64(2.0 * (Float64(Float64(l / k) * Float64((t_m ^ -0.5) / k)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(Float64(t_m * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 2.2e-8)
		tmp = 2.0 * (((l / k) * ((t_m ^ -0.5) / k)) ^ 2.0);
	else
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_m * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 2.2e-8], N[(2.0 * N[Power[N[(N[(l / k), $MachinePrecision] * N[(N[Power[t$95$m, -0.5], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 2.1999999999999998e-8

   1. Initial program 37.6%

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

      1. associate-/r* 36.9%

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

      2. *-commutative 36.9%

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

      3. associate-*l* 36.9%

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

      4. associate-*l/ 38.0%

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

      5. +-commutative 38.0%

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

      6. unpow2 38.0%

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

      7. sqr-neg 38.0%

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

      8. distribute-frac-neg 38.0%

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

      9. distribute-frac-neg 38.0%

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

      10. unpow2 38.0%

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

      11. associate--l+ 43.6%

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

      12. metadata-eval 43.6%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 43.6%

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

      14. unpow2 43.6%

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

      15. distribute-frac-neg 43.6%

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

      16. distribute-frac-neg 43.6%

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

   3. Simplified 43.6%

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

   4. Taylor expanded in k around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-/r* 62.9%

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

   6. Simplified 62.9%

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

      1. add-sqr-sqrt 31.8%

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

      2. sqrt-div 25.4%

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

      3. sqrt-div 19.4%

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

      4. pow2 19.4%

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

      5. sqrt-prod 9.7%

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

      6. add-sqr-sqrt 13.0%

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

      7. sqrt-pow1 13.0%

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

      8. metadata-eval 13.0%

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

      9. sqrt-div 13.0%

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

      10. sqrt-div 13.1%

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

      11. pow2 13.1%

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

      12. sqrt-prod 10.8%

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

      13. add-sqr-sqrt 24.2%

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

      14. sqrt-pow1 26.8%

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

      15. metadata-eval 26.8%

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

   8. Applied egg-rr 26.8%

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

      1. unpow2 26.8%

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

      2. associate-/l/ 26.9%

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

   10. Simplified 26.9%

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

   11. Taylor expanded in l around 0 27.8%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. associate-*l/ 26.8%

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

   13. Simplified 26.8%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]
   14. Step-by-step derivation

       1. *-commutative 26.8%

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

       2. unpow2 26.8%

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

       3. times-frac 28.8%

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

       4. inv-pow 28.8%

          \[\leadsto 2 \cdot {\left(\frac{\sqrt{\color{blue}{{t}^{-1}}}}{k} \cdot \frac{\ell}{k}\right)}^{2} \]

       5. sqrt-pow1 28.8%

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

       6. metadata-eval 28.8%

          \[\leadsto 2 \cdot {\left(\frac{{t}^{\color{blue}{-0.5}}}{k} \cdot \frac{\ell}{k}\right)}^{2} \]

   15. Applied egg-rr 28.8%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{{t}^{-0.5}}{k} \cdot \frac{\ell}{k}\right)}}^{2} \]

   if 2.1999999999999998e-8 < k

   1. Initial program 34.5%

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

   2. Taylor expanded in t around 0 72.3%

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

      1. times-frac 77.7%

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

   4. Simplified 77.7%

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

      1. add-sqr-sqrt 77.8%

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

      2. sqrt-div 77.8%

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

      3. unpow2 77.8%

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

      4. sqrt-prod 77.7%

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

      5. add-sqr-sqrt 77.8%

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

      6. pow2 77.8%

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

      7. sqrt-prod 34.3%

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

      8. add-sqr-sqrt 58.8%

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

      9. sqrt-div 58.8%

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

      10. unpow2 58.8%

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

      11. sqrt-prod 59.0%

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

      12. add-sqr-sqrt 59.0%

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

      13. pow2 59.0%

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

      14. sqrt-prod 40.1%

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

      15. add-sqr-sqrt 96.7%

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

   6. Applied egg-rr 96.7%

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

      1. unpow2 96.7%

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

      2. sin-mult 96.1%

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \color{blue}{\frac{\cos \left(k - k\right) - \cos \left(k + k\right)}{2}}}{\cos k}} \]

   8. Applied egg-rr 96.1%

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

      1. div-sub 96.1%

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \color{blue}{\left(\frac{\cos \left(k - k\right)}{2} - \frac{\cos \left(k + k\right)}{2}\right)}}{\cos k}} \]

      2. +-inverses 96.1%

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}} \]

      3. cos-0 96.1%

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}} \]

      4. metadata-eval 96.1%

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}} \]

      5. count-2 96.1%

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)}{\cos k}} \]

      6. *-commutative 96.1%

         \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right)}{\cos k}} \]

   10. Simplified 96.1%

       \[\leadsto \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{\cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{{t}^{-0.5}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \]

Alternative 8: 73.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-44}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{{t_m}^{-0.5}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t_m \cdot {k}^{2}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 1.15e-44)
    (* 2.0 (pow (* (/ l k) (/ (pow t_m -0.5) k)) 2.0))
    (/ 2.0 (* (* (/ k l) (/ k l)) (/ (* t_m (pow k 2.0)) (cos k)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e-44) {
		tmp = 2.0 * pow(((l / k) * (pow(t_m, -0.5) / k)), 2.0);
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_m * pow(k, 2.0)) / cos(k)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.15d-44) then
        tmp = 2.0d0 * (((l / k) * ((t_m ** (-0.5d0)) / k)) ** 2.0d0)
    else
        tmp = 2.0d0 / (((k / l) * (k / l)) * ((t_m * (k ** 2.0d0)) / cos(k)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 1.15e-44) {
		tmp = 2.0 * Math.pow(((l / k) * (Math.pow(t_m, -0.5) / k)), 2.0);
	} else {
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_m * Math.pow(k, 2.0)) / Math.cos(k)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if k <= 1.15e-44:
		tmp = 2.0 * math.pow(((l / k) * (math.pow(t_m, -0.5) / k)), 2.0)
	else:
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_m * math.pow(k, 2.0)) / math.cos(k)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 1.15e-44)
		tmp = Float64(2.0 * (Float64(Float64(l / k) * Float64((t_m ^ -0.5) / k)) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * Float64(k / l)) * Float64(Float64(t_m * (k ^ 2.0)) / cos(k))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (k <= 1.15e-44)
		tmp = 2.0 * (((l / k) * ((t_m ^ -0.5) / k)) ^ 2.0);
	else
		tmp = 2.0 / (((k / l) * (k / l)) * ((t_m * (k ^ 2.0)) / cos(k)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 1.15e-44], N[(2.0 * N[Power[N[(N[(l / k), $MachinePrecision] * N[(N[Power[t$95$m, -0.5], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.14999999999999999e-44

   1. Initial program 38.1%

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

      1. associate-/r* 37.3%

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

      2. *-commutative 37.3%

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

      3. associate-*l* 37.3%

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

      4. associate-*l/ 38.4%

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

      5. +-commutative 38.4%

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

      6. unpow2 38.4%

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

      7. sqr-neg 38.4%

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

      8. distribute-frac-neg 38.4%

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

      9. distribute-frac-neg 38.4%

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

      10. unpow2 38.4%

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

      11. associate--l+ 43.8%

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

      12. metadata-eval 43.8%

          \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

      13. +-rgt-identity 43.8%

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

      14. unpow2 43.8%

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

      15. distribute-frac-neg 43.8%

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

      16. distribute-frac-neg 43.8%

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

   3. Simplified 43.8%

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

   4. Taylor expanded in k around 0 63.2%

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

      1. *-commutative 63.2%

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

      2. associate-/r* 61.3%

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

   6. Simplified 61.3%

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

      1. add-sqr-sqrt 30.8%

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

      2. sqrt-div 24.0%

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

      3. sqrt-div 18.2%

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

      4. pow2 18.2%

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

      5. sqrt-prod 9.1%

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

      6. add-sqr-sqrt 12.7%

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

      7. sqrt-pow1 12.7%

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

      8. metadata-eval 12.7%

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

      9. sqrt-div 12.7%

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

      10. sqrt-div 12.7%

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

      11. pow2 12.7%

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

      12. sqrt-prod 10.2%

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

      13. add-sqr-sqrt 23.4%

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

      14. sqrt-pow1 26.1%

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

      15. metadata-eval 26.1%

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

   8. Applied egg-rr 26.1%

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

      1. unpow2 26.1%

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

      2. associate-/l/ 26.2%

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

   10. Simplified 26.2%

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

   11. Taylor expanded in l around 0 27.2%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
   12. Step-by-step derivation

       1. associate-*l/ 26.1%

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

   13. Simplified 26.1%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]
   14. Step-by-step derivation

       1. *-commutative 26.1%

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

       2. unpow2 26.1%

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

       3. times-frac 28.2%

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

       4. inv-pow 28.2%

          \[\leadsto 2 \cdot {\left(\frac{\sqrt{\color{blue}{{t}^{-1}}}}{k} \cdot \frac{\ell}{k}\right)}^{2} \]

       5. sqrt-pow1 28.2%

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

       6. metadata-eval 28.2%

          \[\leadsto 2 \cdot {\left(\frac{{t}^{\color{blue}{-0.5}}}{k} \cdot \frac{\ell}{k}\right)}^{2} \]

   15. Applied egg-rr 28.2%

       \[\leadsto 2 \cdot {\color{blue}{\left(\frac{{t}^{-0.5}}{k} \cdot \frac{\ell}{k}\right)}}^{2} \]

   if 1.14999999999999999e-44 < k

   1. Initial program 33.9%

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

   2. Taylor expanded in t around 0 75.8%

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

      1. times-frac 80.5%

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

   4. Simplified 80.5%

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

      1. add-sqr-sqrt 80.6%

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

      2. sqrt-div 80.6%

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

      3. unpow2 80.6%

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

      4. sqrt-prod 80.5%

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

      5. add-sqr-sqrt 80.6%

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

      6. pow2 80.6%

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

      7. sqrt-prod 37.6%

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

      8. add-sqr-sqrt 59.0%

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

      9. sqrt-div 59.0%

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

      10. unpow2 59.0%

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

      11. sqrt-prod 59.1%

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

      12. add-sqr-sqrt 59.1%

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

      13. pow2 59.1%

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

      14. sqrt-prod 42.6%

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

      15. add-sqr-sqrt 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in k around 0 63.1%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{-44}:\\ \;\;\;\;2 \cdot {\left(\frac{\ell}{k} \cdot \frac{{t}^{-0.5}}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot {k}^{2}}{\cos k}}\\ \end{array} \]

Alternative 9: 73.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot {\left(\frac{\ell}{k} \cdot \frac{{t_m}^{-0.5}}{k}\right)}^{2}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (pow (* (/ l k) (/ (pow t_m -0.5) k)) 2.0))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * pow(((l / k) * (pow(t_m, -0.5) / k)), 2.0));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l / k) * ((t_m ** (-0.5d0)) / k)) ** 2.0d0))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * Math.pow(((l / k) * (Math.pow(t_m, -0.5) / k)), 2.0));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * math.pow(((l / k) * (math.pow(t_m, -0.5) / k)), 2.0))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * (Float64(Float64(l / k) * Float64((t_m ^ -0.5) / k)) ^ 2.0)))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l / k) * ((t_m ^ -0.5) / k)) ^ 2.0));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[Power[N[(N[(l / k), $MachinePrecision] * N[(N[Power[t$95$m, -0.5], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

   1. associate-/r* 36.3%

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

   2. *-commutative 36.3%

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

   3. associate-*l* 36.3%

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

   4. associate-*l/ 37.1%

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

   5. +-commutative 37.1%

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

   6. unpow2 37.1%

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

   7. sqr-neg 37.1%

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

   8. distribute-frac-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. unpow2 37.1%

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

   11. associate--l+ 44.0%

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

   12. metadata-eval 44.0%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   13. +-rgt-identity 44.0%

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

   14. unpow2 44.0%

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

   15. distribute-frac-neg 44.0%

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

   16. distribute-frac-neg 44.0%

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

3. Simplified 44.0%

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

4. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/r* 60.4%

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

6. Simplified 60.4%

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

   1. add-sqr-sqrt 36.7%

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

   2. sqrt-div 29.0%

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

   3. sqrt-div 21.1%

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

   4. pow2 21.1%

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

   5. sqrt-prod 10.7%

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

   6. add-sqr-sqrt 16.4%

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

   7. sqrt-pow1 16.4%

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

   8. metadata-eval 16.4%

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

   9. sqrt-div 16.4%

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

   10. sqrt-div 16.1%

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

   11. pow2 16.1%

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

   12. sqrt-prod 11.5%

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

   13. add-sqr-sqrt 24.4%

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

   14. sqrt-pow1 26.3%

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

   15. metadata-eval 26.3%

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

8. Applied egg-rr 26.3%

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

   1. unpow2 26.3%

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

   2. associate-/l/ 26.4%

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

10. Simplified 26.4%

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

11. Taylor expanded in l around 0 27.1%

    \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
12. Step-by-step derivation

    1. associate-*l/ 26.3%

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

13. Simplified 26.3%

    \[\leadsto 2 \cdot {\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{t}}}{{k}^{2}}\right)}}^{2} \]
14. Step-by-step derivation

    1. *-commutative 26.3%

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

    2. unpow2 26.3%

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

    3. times-frac 27.9%

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

    4. inv-pow 27.9%

       \[\leadsto 2 \cdot {\left(\frac{\sqrt{\color{blue}{{t}^{-1}}}}{k} \cdot \frac{\ell}{k}\right)}^{2} \]

    5. sqrt-pow1 27.9%

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

    6. metadata-eval 27.9%

       \[\leadsto 2 \cdot {\left(\frac{{t}^{\color{blue}{-0.5}}}{k} \cdot \frac{\ell}{k}\right)}^{2} \]

15. Applied egg-rr 27.9%

    \[\leadsto 2 \cdot {\color{blue}{\left(\frac{{t}^{-0.5}}{k} \cdot \frac{\ell}{k}\right)}}^{2} \]

16. Final simplification 27.9%

    \[\leadsto 2 \cdot {\left(\frac{\ell}{k} \cdot \frac{{t}^{-0.5}}{k}\right)}^{2} \]

Alternative 10: 70.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}{t_m}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (* (/ l (pow k 2.0)) (* (/ l k) (/ 1.0 k))) t_m))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (((l / pow(k, 2.0)) * ((l / k) * (1.0 / k))) / t_m));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * (((l / (k ** 2.0d0)) * ((l / k) * (1.0d0 / k))) / t_m))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * (((l / Math.pow(k, 2.0)) * ((l / k) * (1.0 / k))) / t_m));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * (((l / math.pow(k, 2.0)) * ((l / k) * (1.0 / k))) / t_m))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64(Float64(l / (k ^ 2.0)) * Float64(Float64(l / k) * Float64(1.0 / k))) / t_m)))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * (((l / (k ^ 2.0)) * ((l / k) * (1.0 / k))) / t_m));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

   1. associate-/r* 36.3%

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

   2. *-commutative 36.3%

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

   3. associate-*l* 36.3%

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

   4. associate-*l/ 37.1%

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

   5. +-commutative 37.1%

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

   6. unpow2 37.1%

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

   7. sqr-neg 37.1%

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

   8. distribute-frac-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. unpow2 37.1%

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

   11. associate--l+ 44.0%

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

   12. metadata-eval 44.0%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   13. +-rgt-identity 44.0%

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

   14. unpow2 44.0%

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

   15. distribute-frac-neg 44.0%

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

   16. distribute-frac-neg 44.0%

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

3. Simplified 44.0%

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

4. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/r* 60.4%

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

6. Simplified 60.4%

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

   1. add-sqr-sqrt 36.7%

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

   2. sqrt-div 29.0%

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

   3. sqrt-div 21.1%

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

   4. pow2 21.1%

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

   5. sqrt-prod 10.7%

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

   6. add-sqr-sqrt 16.4%

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

   7. sqrt-pow1 16.4%

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

   8. metadata-eval 16.4%

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

   9. sqrt-div 16.4%

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

   10. sqrt-div 16.1%

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

   11. pow2 16.1%

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

   12. sqrt-prod 11.5%

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

   13. add-sqr-sqrt 24.4%

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

   14. sqrt-pow1 26.3%

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

   15. metadata-eval 26.3%

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

8. Applied egg-rr 26.3%

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

   1. unpow2 26.3%

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

   2. associate-/l/ 26.4%

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

10. Simplified 26.4%

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

    1. unpow2 26.4%

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

    2. associate-/r* 26.4%

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

    3. associate-/r* 27.1%

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

    4. frac-times 27.5%

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

    5. add-sqr-sqrt 72.3%

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

12. Applied egg-rr 72.3%

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

    1. *-un-lft-identity 72.3%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \frac{\color{blue}{1 \cdot \ell}}{{k}^{2}}}{t} \]

    2. unpow2 72.3%

       \[\leadsto 2 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \frac{1 \cdot \ell}{\color{blue}{k \cdot k}}}{t} \]

    3. times-frac 72.3%

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

14. Applied egg-rr 72.3%

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

15. Final simplification 72.3%

    \[\leadsto 2 \cdot \frac{\frac{\ell}{{k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{k}\right)}{t} \]

Alternative 11: 65.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t_m}\right)\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (* (/ l (pow k 4.0)) (/ l t_m)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((l / pow(k, 4.0)) * (l / t_m)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l / (k ** 4.0d0)) * (l / t_m)))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((l / Math.pow(k, 4.0)) * (l / t_m)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((l / math.pow(k, 4.0)) * (l / t_m)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t_m))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l / (k ^ 4.0)) * (l / t_m)));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

   1. associate-/r* 36.3%

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

   2. *-commutative 36.3%

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

   3. associate-*l* 36.3%

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

   4. associate-*l/ 37.1%

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

   5. +-commutative 37.1%

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

   6. unpow2 37.1%

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

   7. sqr-neg 37.1%

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

   8. distribute-frac-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. unpow2 37.1%

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

   11. associate--l+ 44.0%

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

   12. metadata-eval 44.0%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   13. +-rgt-identity 44.0%

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

   14. unpow2 44.0%

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

   15. distribute-frac-neg 44.0%

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

   16. distribute-frac-neg 44.0%

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

3. Simplified 44.0%

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

4. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/r* 60.4%

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

6. Simplified 60.4%

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

   1. pow2 60.4%

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

8. Applied egg-rr 60.4%

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

   1. associate-/l/ 62.1%

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

   2. times-frac 68.4%

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

10. Applied egg-rr 68.4%

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

11. Final simplification 68.4%

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

Alternative 12: 65.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot \frac{\ell \cdot {k}^{-4}}{\frac{t_m}{\ell}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (* l (pow k -4.0)) (/ t_m l)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((l * pow(k, -4.0)) / (t_m / l)));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l * (k ** (-4.0d0))) / (t_m / l)))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((l * Math.pow(k, -4.0)) / (t_m / l)));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((l * math.pow(k, -4.0)) / (t_m / l)))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64(l * (k ^ -4.0)) / Float64(t_m / l))))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l * (k ^ -4.0)) / (t_m / l)));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[(l * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

   1. associate-/r* 36.3%

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

   2. *-commutative 36.3%

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

   3. associate-*l* 36.3%

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

   4. associate-*l/ 37.1%

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

   5. +-commutative 37.1%

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

   6. unpow2 37.1%

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

   7. sqr-neg 37.1%

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

   8. distribute-frac-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. unpow2 37.1%

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

   11. associate--l+ 44.0%

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

   12. metadata-eval 44.0%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   13. +-rgt-identity 44.0%

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

   14. unpow2 44.0%

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

   15. distribute-frac-neg 44.0%

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

   16. distribute-frac-neg 44.0%

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

3. Simplified 44.0%

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

4. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/r* 60.4%

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

6. Simplified 60.4%

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

   1. pow2 60.4%

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

8. Applied egg-rr 60.4%

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

   1. associate-/l/ 62.1%

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

   2. times-frac 68.4%

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

10. Applied egg-rr 68.4%

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

    1. associate-*r/ 69.3%

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

    2. div-inv 69.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \frac{1}{{k}^{4}}\right)} \cdot \ell}{t} \]

    3. pow-flip 69.3%

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

    4. metadata-eval 69.3%

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

12. Applied egg-rr 69.3%

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

    1. associate-/l* 68.8%

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

14. Simplified 68.8%

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

15. Final simplification 68.8%

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

Alternative 13: 66.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \left(2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t_m}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* 2.0 (/ (/ l (/ (pow k 4.0) l)) t_m))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((l / (pow(k, 4.0) / l)) / t_m));
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = t_s * (2.0d0 * ((l / ((k ** 4.0d0) / l)) / t_m))
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	return t_s * (2.0 * ((l / (Math.pow(k, 4.0) / l)) / t_m));
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	return t_s * (2.0 * ((l / (math.pow(k, 4.0) / l)) / t_m))

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	return Float64(t_s * Float64(2.0 * Float64(Float64(l / Float64((k ^ 4.0) / l)) / t_m)))
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k)
	tmp = t_s * (2.0 * ((l / ((k ^ 4.0) / l)) / t_m));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * N[(2.0 * N[(N[(l / N[(N[Power[k, 4.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

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

   1. associate-/r* 36.3%

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

   2. *-commutative 36.3%

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

   3. associate-*l* 36.3%

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

   4. associate-*l/ 37.1%

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

   5. +-commutative 37.1%

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

   6. unpow2 37.1%

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

   7. sqr-neg 37.1%

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

   8. distribute-frac-neg 37.1%

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

   9. distribute-frac-neg 37.1%

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

   10. unpow2 37.1%

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

   11. associate--l+ 44.0%

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

   12. metadata-eval 44.0%

       \[\leadsto \frac{\frac{2}{\sin k \cdot \frac{{t}^{3} \cdot \tan k}{\ell \cdot \ell}}}{{\left(\frac{-k}{t}\right)}^{2} + \color{blue}{0}} \]

   13. +-rgt-identity 44.0%

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

   14. unpow2 44.0%

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

   15. distribute-frac-neg 44.0%

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

   16. distribute-frac-neg 44.0%

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

3. Simplified 44.0%

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

4. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/r* 60.4%

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

6. Simplified 60.4%

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

   1. add-sqr-sqrt 36.7%

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

   2. sqrt-div 29.0%

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

   3. sqrt-div 21.1%

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

   4. pow2 21.1%

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

   5. sqrt-prod 10.7%

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

   6. add-sqr-sqrt 16.4%

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

   7. sqrt-pow1 16.4%

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

   8. metadata-eval 16.4%

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

   9. sqrt-div 16.4%

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

   10. sqrt-div 16.1%

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

   11. pow2 16.1%

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

   12. sqrt-prod 11.5%

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

   13. add-sqr-sqrt 24.4%

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

   14. sqrt-pow1 26.3%

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

   15. metadata-eval 26.3%

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

8. Applied egg-rr 26.3%

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

   1. unpow2 26.3%

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

   2. associate-/l/ 26.4%

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

10. Simplified 26.4%

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

    1. unpow2 26.4%

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

    2. associate-/r* 26.4%

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

    3. associate-/r* 27.1%

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

    4. frac-times 27.5%

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

    5. add-sqr-sqrt 72.3%

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

12. Applied egg-rr 72.3%

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

    1. clear-num 72.3%

       \[\leadsto 2 \cdot \frac{\color{blue}{\frac{1}{\frac{{k}^{2}}{\ell}}} \cdot \frac{\ell}{{k}^{2}}}{t} \]

    2. unpow2 72.3%

       \[\leadsto 2 \cdot \frac{\frac{1}{\frac{\color{blue}{k \cdot k}}{\ell}} \cdot \frac{\ell}{{k}^{2}}}{t} \]

    3. associate-*l/ 72.3%

       \[\leadsto 2 \cdot \frac{\frac{1}{\color{blue}{\frac{k}{\ell} \cdot k}} \cdot \frac{\ell}{{k}^{2}}}{t} \]

    4. frac-times 71.6%

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

    5. *-un-lft-identity 71.6%

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

    6. associate-*l/ 71.6%

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

    7. unpow2 71.6%

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

14. Applied egg-rr 71.6%

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

    1. associate-*l/ 69.3%

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

    2. pow-sqr 69.3%

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

    3. metadata-eval 69.3%

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

16. Simplified 69.3%

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

17. Final simplification 69.3%

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

Reproduce

herbie shell --seed 2023318 
(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))))