Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.6% → 99.6%

Time: 19.4s

Alternatives: 9

Speedup: 3.8×

• 34.2% 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: 34.6% 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: 99.6% accurate, 0.8× speedup

Math

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

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
   (*
    (cos k)
    (/
     (/ (/ l k) (sin k))
     (* (sqrt t_m) (* (* (sin k) (sqrt t_m)) (/ k 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 * (cos(k) * (((l / k) / sin(k)) / (sqrt(t_m) * ((sin(k) * sqrt(t_m)) * (k / 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 * (cos(k) * (((l / k) / sin(k)) / (sqrt(t_m) * ((sin(k) * sqrt(t_m)) * (k / 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 * (Math.cos(k) * (((l / k) / Math.sin(k)) / (Math.sqrt(t_m) * ((Math.sin(k) * Math.sqrt(t_m)) * (k / 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 * (math.cos(k) * (((l / k) / math.sin(k)) / (math.sqrt(t_m) * ((math.sin(k) * math.sqrt(t_m)) * (k / 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(cos(k) * Float64(Float64(Float64(l / k) / sin(k)) / Float64(sqrt(t_m) * Float64(Float64(sin(k) * sqrt(t_m)) * Float64(k / 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 * (cos(k) * (((l / k) / sin(k)) / (sqrt(t_m) * ((sin(k) * sqrt(t_m)) * (k / 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[Cos[k], $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

   1. associate-/r* 37.4%

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

   2. *-commutative 37.4%

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

   3. associate-*l* 37.4%

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

   4. associate-*l/ 37.2%

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

   5. +-commutative 37.2%

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

   6. unpow2 37.2%

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

   7. sqr-neg 37.2%

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

   8. distribute-frac-neg 37.2%

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

   9. distribute-frac-neg 37.2%

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

   10. unpow2 37.2%

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

   11. associate--l+ 45.0%

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

   12. metadata-eval 45.0%

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

   13. +-rgt-identity 45.0%

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

   14. unpow2 45.0%

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

   15. distribute-frac-neg 45.0%

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

   16. distribute-frac-neg 45.0%

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

3. Simplified 45.0%

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

4. Taylor expanded in k around inf 72.8%

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

   1. associate-/l* 72.8%

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

6. Simplified 72.8%

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

   1. expm1-log1p-u 52.7%

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

   2. expm1-udef 28.2%

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

8. Applied egg-rr 29.0%

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

   1. expm1-def 31.7%

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

   2. expm1-log1p 32.2%

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

10. Simplified 32.2%

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

    1. expm1-log1p-u 30.5%

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

    2. expm1-udef 25.8%

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

12. Applied egg-rr 29.1%

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

    1. expm1-def 36.7%

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

    2. expm1-log1p 40.0%

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

    3. *-commutative 40.0%

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

    4. associate-/r* 40.8%

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

14. Simplified 40.8%

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

    1. unpow2 40.8%

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

    2. clear-num 40.8%

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

    3. associate-/r* 40.8%

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

    4. frac-times 40.9%

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

    5. *-un-lft-identity 40.9%

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

    6. div-inv 40.8%

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

    7. clear-num 40.8%

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

16. Applied egg-rr 40.8%

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

17. Final simplification 40.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

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 (* (cos k) (pow (/ (/ l k) (* (sin k) (sqrt t_m))) 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 * (cos(k) * pow(((l / k) / (sin(k) * sqrt(t_m))), 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 * (cos(k) * (((l / k) / (sin(k) * sqrt(t_m))) ** 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.cos(k) * Math.pow(((l / k) / (Math.sin(k) * Math.sqrt(t_m))), 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.cos(k) * math.pow(((l / k) / (math.sin(k) * math.sqrt(t_m))), 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(cos(k) * (Float64(Float64(l / k) / Float64(sin(k) * sqrt(t_m))) ^ 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 * (cos(k) * (((l / k) / (sin(k) * sqrt(t_m))) ^ 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[(N[Cos[k], $MachinePrecision] * N[Power[N[(N[(l / k), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

   1. associate-/r* 37.4%

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

   2. *-commutative 37.4%

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

   3. associate-*l* 37.4%

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

   4. associate-*l/ 37.2%

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

   5. +-commutative 37.2%

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

   6. unpow2 37.2%

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

   7. sqr-neg 37.2%

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

   8. distribute-frac-neg 37.2%

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

   9. distribute-frac-neg 37.2%

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

   10. unpow2 37.2%

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

   11. associate--l+ 45.0%

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

   12. metadata-eval 45.0%

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

   13. +-rgt-identity 45.0%

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

   14. unpow2 45.0%

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

   15. distribute-frac-neg 45.0%

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

   16. distribute-frac-neg 45.0%

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

3. Simplified 45.0%

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

4. Taylor expanded in k around inf 72.8%

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

   1. associate-/l* 72.8%

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

6. Simplified 72.8%

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

   1. expm1-log1p-u 52.7%

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

   2. expm1-udef 28.2%

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

8. Applied egg-rr 29.0%

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

   1. expm1-def 31.7%

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

   2. expm1-log1p 32.2%

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

10. Simplified 32.2%

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

    1. expm1-log1p-u 30.5%

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

    2. expm1-udef 25.8%

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

12. Applied egg-rr 29.1%

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

    1. expm1-def 36.7%

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

    2. expm1-log1p 40.0%

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

    3. *-commutative 40.0%

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

    4. associate-/r* 40.8%

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

14. Simplified 40.8%

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

15. Final simplification 40.8%

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

Alternative 3: 85.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\frac{\ell}{k}}{\sin k}\\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot {\left(\frac{\frac{\ell}{k}}{k \cdot \sqrt{t_m}}\right)}^{2}\right)\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+106}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{t_m \cdot {\left(k \cdot \sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{t_2 \cdot t_2}{t_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ (/ l k) (sin k))))
   (*
    t_s
    (if (<= k 4.6e-23)
      (* 2.0 (* (cos k) (pow (/ (/ l k) (* k (sqrt t_m))) 2.0)))
      (if (<= k 3.1e+106)
        (* 2.0 (* (cos k) (* l (/ l (* t_m (pow (* k (sin k)) 2.0))))))
        (* 2.0 (* (cos k) (/ (* t_2 t_2) 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) {
	double t_2 = (l / k) / sin(k);
	double tmp;
	if (k <= 4.6e-23) {
		tmp = 2.0 * (cos(k) * pow(((l / k) / (k * sqrt(t_m))), 2.0));
	} else if (k <= 3.1e+106) {
		tmp = 2.0 * (cos(k) * (l * (l / (t_m * pow((k * sin(k)), 2.0)))));
	} else {
		tmp = 2.0 * (cos(k) * ((t_2 * t_2) / t_m));
	}
	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) :: t_2
    real(8) :: tmp
    t_2 = (l / k) / sin(k)
    if (k <= 4.6d-23) then
        tmp = 2.0d0 * (cos(k) * (((l / k) / (k * sqrt(t_m))) ** 2.0d0))
    else if (k <= 3.1d+106) then
        tmp = 2.0d0 * (cos(k) * (l * (l / (t_m * ((k * sin(k)) ** 2.0d0)))))
    else
        tmp = 2.0d0 * (cos(k) * ((t_2 * t_2) / t_m))
    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 t_2 = (l / k) / Math.sin(k);
	double tmp;
	if (k <= 4.6e-23) {
		tmp = 2.0 * (Math.cos(k) * Math.pow(((l / k) / (k * Math.sqrt(t_m))), 2.0));
	} else if (k <= 3.1e+106) {
		tmp = 2.0 * (Math.cos(k) * (l * (l / (t_m * Math.pow((k * Math.sin(k)), 2.0)))));
	} else {
		tmp = 2.0 * (Math.cos(k) * ((t_2 * t_2) / t_m));
	}
	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):
	t_2 = (l / k) / math.sin(k)
	tmp = 0
	if k <= 4.6e-23:
		tmp = 2.0 * (math.cos(k) * math.pow(((l / k) / (k * math.sqrt(t_m))), 2.0))
	elif k <= 3.1e+106:
		tmp = 2.0 * (math.cos(k) * (l * (l / (t_m * math.pow((k * math.sin(k)), 2.0)))))
	else:
		tmp = 2.0 * (math.cos(k) * ((t_2 * t_2) / t_m))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(Float64(l / k) / sin(k))
	tmp = 0.0
	if (k <= 4.6e-23)
		tmp = Float64(2.0 * Float64(cos(k) * (Float64(Float64(l / k) / Float64(k * sqrt(t_m))) ^ 2.0)));
	elseif (k <= 3.1e+106)
		tmp = Float64(2.0 * Float64(cos(k) * Float64(l * Float64(l / Float64(t_m * (Float64(k * sin(k)) ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64(cos(k) * Float64(Float64(t_2 * t_2) / t_m)));
	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)
	t_2 = (l / k) / sin(k);
	tmp = 0.0;
	if (k <= 4.6e-23)
		tmp = 2.0 * (cos(k) * (((l / k) / (k * sqrt(t_m))) ^ 2.0));
	elseif (k <= 3.1e+106)
		tmp = 2.0 * (cos(k) * (l * (l / (t_m * ((k * sin(k)) ^ 2.0)))));
	else
		tmp = 2.0 * (cos(k) * ((t_2 * t_2) / t_m));
	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_] := Block[{t$95$2 = N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k, 4.6e-23], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[Power[N[(N[(l / k), $MachinePrecision] / N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.1e+106], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(l * N[(l / N[(t$95$m * N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(t$95$2 * t$95$2), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < 4.6000000000000002e-23

   1. Initial program 36.2%

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

      1. associate-/r* 36.2%

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

      2. *-commutative 36.2%

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

      3. associate-*l* 36.2%

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

      4. associate-*l/ 35.8%

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

      5. +-commutative 35.8%

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

      6. unpow2 35.8%

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

      7. sqr-neg 35.8%

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

      8. distribute-frac-neg 35.8%

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

      9. distribute-frac-neg 35.8%

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

      10. unpow2 35.8%

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

      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 72.5%

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

      1. associate-/l* 72.5%

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

   6. Simplified 72.5%

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

      1. expm1-log1p-u 59.7%

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

      2. expm1-udef 27.8%

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

   8. Applied egg-rr 29.2%

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

      1. expm1-def 32.1%

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

      2. expm1-log1p 32.7%

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

   10. Simplified 32.7%

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

       1. expm1-log1p-u 31.4%

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

       2. expm1-udef 25.5%

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

   12. Applied egg-rr 29.3%

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

       1. expm1-def 38.5%

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

       2. expm1-log1p 41.1%

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

       3. *-commutative 41.1%

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

       4. associate-/r* 41.7%

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

   14. Simplified 41.7%

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

   15. Taylor expanded in k around 0 28.7%

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

   if 4.6000000000000002e-23 < k < 3.0999999999999999e106

   1. Initial program 34.0%

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

      1. associate-/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/ 36.3%

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

      5. +-commutative 36.3%

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

      6. unpow2 36.3%

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

      7. sqr-neg 36.3%

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

      8. distribute-frac-neg 36.3%

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

      9. distribute-frac-neg 36.3%

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

      10. unpow2 36.3%

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

      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 inf 96.6%

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

      1. associate-/l* 96.7%

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

   6. Simplified 96.7%

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

      1. expm1-log1p-u 50.8%

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

      2. expm1-udef 19.4%

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

   8. Applied egg-rr 12.2%

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

      1. expm1-def 19.1%

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

      2. expm1-log1p 19.1%

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

   10. Simplified 19.1%

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

       1. associate-/r/ 19.1%

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

       2. associate-*r* 19.0%

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

       3. unpow-prod-down 19.1%

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

       4. pow2 19.1%

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

       5. add-sqr-sqrt 96.6%

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

   12. Applied egg-rr 96.6%

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

       1. unpow2 96.6%

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

       2. *-un-lft-identity 96.6%

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

       3. times-frac 99.4%

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

       4. *-commutative 99.4%

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

   14. Applied egg-rr 99.4%

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

   if 3.0999999999999999e106 < k

   1. Initial program 41.5%

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

      1. associate-/r* 41.5%

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

      2. *-commutative 41.5%

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

      3. associate-*l* 41.5%

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

      4. associate-*l/ 41.5%

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

      5. +-commutative 41.5%

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

      6. unpow2 41.5%

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

      7. sqr-neg 41.5%

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

      8. distribute-frac-neg 41.5%

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

      9. distribute-frac-neg 41.5%

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

      10. unpow2 41.5%

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

      11. associate--l+ 50.2%

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

      12. metadata-eval 50.2%

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

      13. +-rgt-identity 50.2%

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

      14. unpow2 50.2%

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

      15. distribute-frac-neg 50.2%

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

      16. distribute-frac-neg 50.2%

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

   3. Simplified 50.2%

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

   4. Taylor expanded in k around inf 62.8%

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

      1. associate-/l* 62.8%

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

   6. Simplified 62.8%

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

      1. expm1-log1p-u 33.1%

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

      2. expm1-udef 33.1%

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

   8. Applied egg-rr 36.1%

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

      1. expm1-def 36.1%

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

      2. expm1-log1p 36.3%

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

   10. Simplified 36.3%

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

       1. expm1-log1p-u 36.3%

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

       2. expm1-udef 33.1%

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

   12. Applied egg-rr 36.5%

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

       1. expm1-def 43.0%

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

       2. expm1-log1p 46.4%

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

       3. *-commutative 46.4%

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

       4. associate-/r* 48.1%

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

   14. Simplified 48.1%

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

       1. unpow2 48.1%

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

       2. associate-/r* 48.0%

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

       3. associate-/r* 48.0%

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

       4. frac-times 46.4%

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

       5. add-sqr-sqrt 94.6%

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

   16. Applied egg-rr 94.6%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot {\left(\frac{\frac{\ell}{k}}{k \cdot \sqrt{t}}\right)}^{2}\right)\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+106}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \left(\ell \cdot \frac{\ell}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{\ell}{k}}{\sin k}}{t}\right)\\ \end{array} \]

Alternative 4: 81.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot {\left(\frac{\frac{\ell}{k}}{k \cdot \sqrt{t_m}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \left(\frac{\ell}{{\left(k \cdot \sin k\right)}^{2}} \cdot \frac{\ell}{t_m}\right)\right)\\ \end{array} \end{array} \]

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 1.1e-37)
    (* 2.0 (* (cos k) (pow (/ (/ l k) (* k (sqrt t_m))) 2.0)))
    (* 2.0 (* (cos k) (* (/ l (pow (* k (sin k)) 2.0)) (/ 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) {
	double tmp;
	if (k <= 1.1e-37) {
		tmp = 2.0 * (cos(k) * pow(((l / k) / (k * sqrt(t_m))), 2.0));
	} else {
		tmp = 2.0 * (cos(k) * ((l / pow((k * sin(k)), 2.0)) * (l / t_m)));
	}
	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 <= 1.1d-37) then
        tmp = 2.0d0 * (cos(k) * (((l / k) / (k * sqrt(t_m))) ** 2.0d0))
    else
        tmp = 2.0d0 * (cos(k) * ((l / ((k * sin(k)) ** 2.0d0)) * (l / t_m)))
    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 <= 1.1e-37) {
		tmp = 2.0 * (Math.cos(k) * Math.pow(((l / k) / (k * Math.sqrt(t_m))), 2.0));
	} else {
		tmp = 2.0 * (Math.cos(k) * ((l / Math.pow((k * Math.sin(k)), 2.0)) * (l / t_m)));
	}
	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 <= 1.1e-37:
		tmp = 2.0 * (math.cos(k) * math.pow(((l / k) / (k * math.sqrt(t_m))), 2.0))
	else:
		tmp = 2.0 * (math.cos(k) * ((l / math.pow((k * math.sin(k)), 2.0)) * (l / t_m)))
	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 <= 1.1e-37)
		tmp = Float64(2.0 * Float64(cos(k) * (Float64(Float64(l / k) / Float64(k * sqrt(t_m))) ^ 2.0)));
	else
		tmp = Float64(2.0 * Float64(cos(k) * Float64(Float64(l / (Float64(k * sin(k)) ^ 2.0)) * Float64(l / t_m))));
	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 <= 1.1e-37)
		tmp = 2.0 * (cos(k) * (((l / k) / (k * sqrt(t_m))) ^ 2.0));
	else
		tmp = 2.0 * (cos(k) * ((l / ((k * sin(k)) ^ 2.0)) * (l / t_m)));
	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, 1.1e-37], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[Power[N[(N[(l / k), $MachinePrecision] / N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[(l / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000001e-37

   1. Initial program 37.0%

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

      1. associate-/r* 37.0%

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

      2. *-commutative 37.0%

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

      3. associate-*l* 37.0%

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

      4. associate-*l/ 36.7%

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

      5. +-commutative 36.7%

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

      6. unpow2 36.7%

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

      7. sqr-neg 36.7%

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

      8. distribute-frac-neg 36.7%

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

      9. distribute-frac-neg 36.7%

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

      10. unpow2 36.7%

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

      11. associate--l+ 43.3%

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

      12. metadata-eval 43.3%

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

      13. +-rgt-identity 43.3%

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

      14. unpow2 43.3%

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

      15. distribute-frac-neg 43.3%

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

      16. distribute-frac-neg 43.3%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in k around inf 72.5%

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

      1. associate-/l* 72.5%

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

   6. Simplified 72.5%

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

      1. expm1-log1p-u 59.3%

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

      2. expm1-udef 27.3%

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

   8. Applied egg-rr 29.3%

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

      1. expm1-def 31.7%

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

      2. expm1-log1p 32.3%

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

   10. Simplified 32.3%

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

       1. expm1-log1p-u 31.0%

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

       2. expm1-udef 24.9%

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

   12. Applied egg-rr 28.8%

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

       1. expm1-def 38.2%

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

       2. expm1-log1p 40.9%

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

       3. *-commutative 40.9%

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

       4. associate-/r* 41.5%

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

   14. Simplified 41.5%

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

   15. Taylor expanded in k around 0 28.2%

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

   if 1.10000000000000001e-37 < k

   1. Initial program 37.4%

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

      1. associate-/r* 38.1%

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

      2. *-commutative 38.1%

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

      3. associate-*l* 38.1%

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

      4. associate-*l/ 38.1%

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

      5. +-commutative 38.1%

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

      6. unpow2 38.1%

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

      7. sqr-neg 38.1%

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

      8. distribute-frac-neg 38.1%

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

      9. distribute-frac-neg 38.1%

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

      10. unpow2 38.1%

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

      11. associate--l+ 48.4%

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

      12. metadata-eval 48.4%

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

      13. +-rgt-identity 48.4%

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

      14. unpow2 48.4%

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

      15. distribute-frac-neg 48.4%

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

      16. distribute-frac-neg 48.4%

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

   3. Simplified 48.4%

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

   4. Taylor expanded in k around inf 73.3%

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

      1. associate-/l* 73.4%

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

   6. Simplified 73.4%

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

      1. expm1-log1p-u 40.2%

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

      2. expm1-udef 29.8%

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

   8. Applied egg-rr 28.5%

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

      1. expm1-def 31.7%

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

      2. expm1-log1p 31.8%

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

   10. Simplified 31.8%

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

       1. associate-/r/ 31.8%

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

       2. associate-*r* 31.8%

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

       3. unpow-prod-down 29.7%

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

       4. pow2 29.7%

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

       5. add-sqr-sqrt 73.3%

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

   12. Applied egg-rr 73.3%

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

       1. unpow2 73.3%

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

       2. times-frac 77.6%

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

   14. Applied egg-rr 77.6%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot {\left(\frac{\frac{\ell}{k}}{k \cdot \sqrt{t}}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \left(\frac{\ell}{{\left(k \cdot \sin k\right)}^{2}} \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \]

Alternative 5: 74.6% accurate, 1.3× speedup

Math

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

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 (* (cos k) (pow (/ (/ l k) (* k (sqrt t_m))) 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 * (cos(k) * pow(((l / k) / (k * sqrt(t_m))), 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 * (cos(k) * (((l / k) / (k * sqrt(t_m))) ** 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.cos(k) * Math.pow(((l / k) / (k * Math.sqrt(t_m))), 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.cos(k) * math.pow(((l / k) / (k * math.sqrt(t_m))), 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(cos(k) * (Float64(Float64(l / k) / Float64(k * sqrt(t_m))) ^ 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 * (cos(k) * (((l / k) / (k * sqrt(t_m))) ^ 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[(N[Cos[k], $MachinePrecision] * N[Power[N[(N[(l / k), $MachinePrecision] / N[(k * N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

   1. associate-/r* 37.4%

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

   2. *-commutative 37.4%

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

   3. associate-*l* 37.4%

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

   4. associate-*l/ 37.2%

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

   5. +-commutative 37.2%

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

   6. unpow2 37.2%

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

   7. sqr-neg 37.2%

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

   8. distribute-frac-neg 37.2%

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

   9. distribute-frac-neg 37.2%

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

   10. unpow2 37.2%

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

   11. associate--l+ 45.0%

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

   12. metadata-eval 45.0%

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

   13. +-rgt-identity 45.0%

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

   14. unpow2 45.0%

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

   15. distribute-frac-neg 45.0%

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

   16. distribute-frac-neg 45.0%

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

3. Simplified 45.0%

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

4. Taylor expanded in k around inf 72.8%

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

   1. associate-/l* 72.8%

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

6. Simplified 72.8%

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

   1. expm1-log1p-u 52.7%

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

   2. expm1-udef 28.2%

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

8. Applied egg-rr 29.0%

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

   1. expm1-def 31.7%

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

   2. expm1-log1p 32.2%

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

10. Simplified 32.2%

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

    1. expm1-log1p-u 30.5%

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

    2. expm1-udef 25.8%

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

12. Applied egg-rr 29.1%

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

    1. expm1-def 36.7%

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

    2. expm1-log1p 40.0%

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

    3. *-commutative 40.0%

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

    4. associate-/r* 40.8%

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

14. Simplified 40.8%

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

15. Taylor expanded in k around 0 28.4%

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

16. Final simplification 28.4%

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

Alternative 6: 71.3% accurate, 2.0× speedup

Math

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

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 2.0) l) (/ t_m (* l (pow 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(k, 2.0) / l) * (t_m / (l * pow(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 / (((k ** 2.0d0) / l) * (t_m / (l * (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(k, 2.0) / l) * (t_m / (l * Math.pow(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(k, 2.0) / l) * (t_m / (l * math.pow(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((k ^ 2.0) / l) * Float64(t_m / Float64(l * (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 / (((k ^ 2.0) / l) * (t_m / (l * (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[(N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

   1. associate-*l* 37.2%

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

   2. associate-*l/ 36.9%

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

   3. associate--l+ 36.9%

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

3. Simplified 36.9%

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

4. Taylor expanded in k around 0 60.9%

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

   1. expm1-log1p-u 43.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)\right)}} \]

   2. div-inv 42.9%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left({k}^{4} \cdot t\right) \cdot \frac{1}{{\ell}^{2}}}\right)\right)} \]

   3. *-commutative 42.9%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(t \cdot {k}^{4}\right)} \cdot \frac{1}{{\ell}^{2}}\right)\right)} \]

   4. pow-flip 42.9%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot \color{blue}{{\ell}^{\left(-2\right)}}\right)\right)} \]

   5. metadata-eval 42.9%

      \[\leadsto \frac{2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot {\ell}^{\color{blue}{-2}}\right)\right)} \]

6. Applied egg-rr 42.9%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(t \cdot {k}^{4}\right) \cdot {\ell}^{-2}\right)\right)}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 60.2%

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

   2. associate-*l* 60.3%

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

   3. metadata-eval 60.3%

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

   4. pow-prod-up 60.3%

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

   5. add-sqr-sqrt 60.3%

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

   6. unswap-sqr 61.2%

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

   7. sqrt-pow1 49.0%

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

   8. metadata-eval 49.0%

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

   9. unpow-1 49.0%

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

   10. div-inv 49.0%

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

   11. sqrt-pow1 70.9%

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

   12. metadata-eval 70.9%

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

   13. unpow-1 70.9%

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

   14. div-inv 70.9%

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

   15. times-frac 61.0%

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

   16. pow-sqr 61.0%

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

   17. metadata-eval 61.0%

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

   18. unpow2 61.0%

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

   19. clear-num 61.0%

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

   20. div-inv 61.0%

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

   21. *-un-lft-identity 61.0%

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

8. Applied egg-rr 72.9%

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

9. Final simplification 72.9%

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

Alternative 7: 69.7% accurate, 2.0× speedup

Math

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

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 (/ (pow (* l (pow k -2.0)) 2.0) 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 * (pow((l * pow(k, -2.0)), 2.0) / 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))) ** 2.0d0) / 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 * (Math.pow((l * Math.pow(k, -2.0)), 2.0) / 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 * (math.pow((l * math.pow(k, -2.0)), 2.0) / 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 ^ -2.0)) ^ 2.0) / 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)) ^ 2.0) / 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[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

   1. associate-/r* 37.4%

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

   2. *-commutative 37.4%

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

   3. associate-*l* 37.4%

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

   4. associate-*l/ 37.2%

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

   5. +-commutative 37.2%

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

   6. unpow2 37.2%

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

   7. sqr-neg 37.2%

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

   8. distribute-frac-neg 37.2%

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

   9. distribute-frac-neg 37.2%

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

   10. unpow2 37.2%

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

   11. associate--l+ 45.0%

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

   12. metadata-eval 45.0%

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

   13. +-rgt-identity 45.0%

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

   14. unpow2 45.0%

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

   15. distribute-frac-neg 45.0%

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

   16. distribute-frac-neg 45.0%

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

3. Simplified 45.0%

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

4. Taylor expanded in k around inf 72.8%

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

   1. associate-/l* 72.8%

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

6. Simplified 72.8%

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

   1. expm1-log1p-u 52.7%

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

   2. expm1-udef 28.2%

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

8. Applied egg-rr 29.0%

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

   1. expm1-def 31.7%

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

   2. expm1-log1p 32.2%

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

10. Simplified 32.2%

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

11. Taylor expanded in k around 0 60.9%

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

    1. associate-/r* 61.0%

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

13. Simplified 61.0%

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

    1. expm1-log1p-u 60.9%

       \[\leadsto 2 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)\right)}}{t} \]

    2. expm1-udef 60.6%

       \[\leadsto 2 \cdot \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{{\ell}^{2}}{{k}^{4}}\right)} - 1}}{t} \]

    3. add-sqr-sqrt 60.6%

       \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{{\ell}^{2}}{{k}^{4}}} \cdot \sqrt{\frac{{\ell}^{2}}{{k}^{4}}}}\right)} - 1}{t} \]

    4. add-sqr-sqrt 60.6%

       \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{{\ell}^{2}}{{k}^{4}}}\right)} - 1}{t} \]

    5. unpow2 60.6%

       \[\leadsto 2 \cdot \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{4}}\right)} - 1}{t} \]

    6. metadata-eval 60.6%

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

    7. pow-prod-up 60.6%

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

    8. frac-times 67.4%

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

    9. pow2 67.4%

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

    10. div-inv 67.4%

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

    11. pow-flip 67.4%

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

    12. metadata-eval 67.4%

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

15. Applied egg-rr 67.4%

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

    1. expm1-def 70.7%

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

    2. expm1-log1p 70.9%

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

17. Simplified 70.9%

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

18. Final simplification 70.9%

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

Alternative 8: 65.5% accurate, 3.8× speedup

Math

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

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 t_m) (/ l (pow k 4.0))))))

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 * ((l / t_m) * (l / pow(k, 4.0))));
}

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 / t_m) * (l / (k ** 4.0d0))))
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 * ((l / t_m) * (l / Math.pow(k, 4.0))));
}

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 * ((l / t_m) * (l / math.pow(k, 4.0))))

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 / t_m) * Float64(l / (k ^ 4.0)))))
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 / t_m) * (l / (k ^ 4.0))));
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[(N[(l / t$95$m), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

   1. associate-/r* 37.4%

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

   2. *-commutative 37.4%

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

   3. associate-*l* 37.4%

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

   4. associate-*l/ 37.2%

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

   5. +-commutative 37.2%

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

   6. unpow2 37.2%

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

   7. sqr-neg 37.2%

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

   8. distribute-frac-neg 37.2%

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

   9. distribute-frac-neg 37.2%

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

   10. unpow2 37.2%

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

   11. associate--l+ 45.0%

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

   12. metadata-eval 45.0%

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

   13. +-rgt-identity 45.0%

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

   14. unpow2 45.0%

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

   15. distribute-frac-neg 45.0%

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

   16. distribute-frac-neg 45.0%

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

3. Simplified 45.0%

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

4. Taylor expanded in k around inf 72.8%

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

   1. associate-/l* 72.8%

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

6. Simplified 72.8%

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

   1. expm1-log1p-u 52.7%

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

   2. expm1-udef 28.2%

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

8. Applied egg-rr 29.0%

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

   1. expm1-def 31.7%

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

   2. expm1-log1p 32.2%

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

10. Simplified 32.2%

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

11. Taylor expanded in k around 0 60.9%

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

    1. associate-/r* 61.0%

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

13. Simplified 61.0%

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

    1. associate-/l/ 60.9%

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

    2. unpow2 60.9%

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

    3. *-commutative 60.9%

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

    4. times-frac 68.6%

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

15. Applied egg-rr 68.6%

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

16. Final simplification 68.6%

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

Alternative 9: 66.2% accurate, 3.8× speedup

Math

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

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 (/ l (pow k 4.0))) 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 * (l / pow(k, 4.0))) / 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 * (l / (k ** 4.0d0))) / 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 * (l / Math.pow(k, 4.0))) / 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 * (l / math.pow(k, 4.0))) / 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(l / (k ^ 4.0))) / 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 * (l / (k ^ 4.0))) / 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[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.2%

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

   1. associate-/r* 37.4%

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

   2. *-commutative 37.4%

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

   3. associate-*l* 37.4%

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

   4. associate-*l/ 37.2%

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

   5. +-commutative 37.2%

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

   6. unpow2 37.2%

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

   7. sqr-neg 37.2%

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

   8. distribute-frac-neg 37.2%

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

   9. distribute-frac-neg 37.2%

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

   10. unpow2 37.2%

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

   11. associate--l+ 45.0%

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

   12. metadata-eval 45.0%

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

   13. +-rgt-identity 45.0%

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

   14. unpow2 45.0%

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

   15. distribute-frac-neg 45.0%

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

   16. distribute-frac-neg 45.0%

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

3. Simplified 45.0%

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

4. Taylor expanded in k around inf 72.8%

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

   1. associate-/l* 72.8%

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

6. Simplified 72.8%

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

   1. expm1-log1p-u 52.7%

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

   2. expm1-udef 28.2%

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

8. Applied egg-rr 29.0%

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

   1. expm1-def 31.7%

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

   2. expm1-log1p 32.2%

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

10. Simplified 32.2%

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

11. Taylor expanded in k around 0 60.9%

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

    1. associate-/r* 61.0%

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

13. Simplified 61.0%

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

    1. unpow2 61.0%

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

    2. *-un-lft-identity 61.0%

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

    3. times-frac 68.6%

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

15. Applied egg-rr 68.6%

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

16. Final simplification 68.6%

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

Reproduce

herbie shell --seed 2023337 
(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))))