Toniolo and Linder, Equation (10-)

Percentage Accurate: 36.1% → 95.0%

Time: 18.1s

Alternatives: 22

Speedup: 3.8×

• 40.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 36.1% 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: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.5 \cdot 10^{-32}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right) \cdot \sqrt{\frac{\cos k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k\_m}\right)}^{2} \cdot \frac{\frac{\cos k\_m \cdot -2}{t\_m}}{-{\sin k\_m}^{2}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= k_m 3.5e-32)
    (pow (* (* (/ l k_m) (/ (sqrt 2.0) k_m)) (sqrt (/ (cos k_m) t_m))) 2.0)
    (*
     (pow (/ l k_m) 2.0)
     (/ (/ (* (cos k_m) -2.0) t_m) (- (pow (sin k_m) 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (k_m <= 3.5e-32) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / k_m)) * sqrt((cos(k_m) / t_m))), 2.0);
	} else {
		tmp = pow((l / k_m), 2.0) * (((cos(k_m) * -2.0) / t_m) / -pow(sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (k_m <= 3.5d-32) then
        tmp = (((l / k_m) * (sqrt(2.0d0) / k_m)) * sqrt((cos(k_m) / t_m))) ** 2.0d0
    else
        tmp = ((l / k_m) ** 2.0d0) * (((cos(k_m) * (-2.0d0)) / t_m) / -(sin(k_m) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (k_m <= 3.5e-32) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / k_m)) * Math.sqrt((Math.cos(k_m) / t_m))), 2.0);
	} else {
		tmp = Math.pow((l / k_m), 2.0) * (((Math.cos(k_m) * -2.0) / t_m) / -Math.pow(Math.sin(k_m), 2.0));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if k_m <= 3.5e-32:
		tmp = math.pow((((l / k_m) * (math.sqrt(2.0) / k_m)) * math.sqrt((math.cos(k_m) / t_m))), 2.0)
	else:
		tmp = math.pow((l / k_m), 2.0) * (((math.cos(k_m) * -2.0) / t_m) / -math.pow(math.sin(k_m), 2.0))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (k_m <= 3.5e-32)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m)) * sqrt(Float64(cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = Float64((Float64(l / k_m) ^ 2.0) * Float64(Float64(Float64(cos(k_m) * -2.0) / t_m) / Float64(-(sin(k_m) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (k_m <= 3.5e-32)
		tmp = (((l / k_m) * (sqrt(2.0) / k_m)) * sqrt((cos(k_m) / t_m))) ^ 2.0;
	else
		tmp = ((l / k_m) ^ 2.0) * (((cos(k_m) * -2.0) / t_m) / -(sin(k_m) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[k$95$m, 3.5e-32], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(N[Power[N[(l / k$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[(N[Cos[k$95$m], $MachinePrecision] * -2.0), $MachinePrecision] / t$95$m), $MachinePrecision] / (-N[Power[N[Sin[k$95$m], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 3.4999999999999999e-32

   1. Initial program 33.7%

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

   3. Simplified 39.8%

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

      1. add-sqr-sqrt 26.9%

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

      2. pow2 26.9%

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

   6. Applied egg-rr 31.4%

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

      1. *-commutative 31.4%

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

   8. Simplified 31.4%

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

   9. Taylor expanded in l around 0 54.8%

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

       1. times-frac 56.0%

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

   11. Simplified 56.0%

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

   12. Taylor expanded in k around 0 47.3%

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

   if 3.4999999999999999e-32 < k

   1. Initial program 26.0%

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

   3. Simplified 36.2%

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

      1. add-sqr-sqrt 33.3%

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

      2. pow2 33.3%

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

   6. Applied egg-rr 18.8%

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

      1. *-commutative 18.8%

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

   8. Simplified 18.8%

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

   9. Taylor expanded in l around 0 55.5%

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

       1. times-frac 55.6%

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

   11. Simplified 55.6%

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

   12. Taylor expanded in t around -inf 0.0%

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

       1. mul-1-neg 0.0%

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

       2. times-frac 0.0%

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

       3. distribute-rgt-neg-in 0.0%

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

       4. unpow2 0.0%

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

       5. unpow2 0.0%

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

       6. times-frac 0.0%

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

       7. unpow2 0.0%

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

       8. associate-/r* 0.0%

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

   14. Simplified 92.0%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-32}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\frac{\cos k \cdot -2}{t}}{-{\sin k}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1}{\frac{k\_m}{t\_m}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.6 \cdot 10^{-69} \lor \neg \left(t\_m \leq 1.25 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(t\_2 \cdot t\_2\right)}{\left(\frac{{t\_m}^{3}}{\ell} \cdot \frac{1}{\ell}\right) \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ 1.0 (/ k_m t_m))))
   (*
    t_s
    (if (or (<= t_m 9.6e-69) (not (<= t_m 1.25e+99)))
      (* (/ (cos k_m) t_m) (pow (* l (/ (sqrt 2.0) (pow k_m 2.0))) 2.0))
      (/
       (* 2.0 (* t_2 t_2))
       (* (* (/ (pow t_m 3.0) l) (/ 1.0 l)) (* (sin k_m) (tan k_m))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = 1.0 / (k_m / t_m);
	double tmp;
	if ((t_m <= 9.6e-69) || !(t_m <= 1.25e+99)) {
		tmp = (cos(k_m) / t_m) * pow((l * (sqrt(2.0) / pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (2.0 * (t_2 * t_2)) / (((pow(t_m, 3.0) / l) * (1.0 / l)) * (sin(k_m) * tan(k_m)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 / (k_m / t_m)
    if ((t_m <= 9.6d-69) .or. (.not. (t_m <= 1.25d+99))) then
        tmp = (cos(k_m) / t_m) * ((l * (sqrt(2.0d0) / (k_m ** 2.0d0))) ** 2.0d0)
    else
        tmp = (2.0d0 * (t_2 * t_2)) / ((((t_m ** 3.0d0) / l) * (1.0d0 / l)) * (sin(k_m) * tan(k_m)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = 1.0 / (k_m / t_m);
	double tmp;
	if ((t_m <= 9.6e-69) || !(t_m <= 1.25e+99)) {
		tmp = (Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / Math.pow(k_m, 2.0))), 2.0);
	} else {
		tmp = (2.0 * (t_2 * t_2)) / (((Math.pow(t_m, 3.0) / l) * (1.0 / l)) * (Math.sin(k_m) * Math.tan(k_m)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = 1.0 / (k_m / t_m)
	tmp = 0
	if (t_m <= 9.6e-69) or not (t_m <= 1.25e+99):
		tmp = (math.cos(k_m) / t_m) * math.pow((l * (math.sqrt(2.0) / math.pow(k_m, 2.0))), 2.0)
	else:
		tmp = (2.0 * (t_2 * t_2)) / (((math.pow(t_m, 3.0) / l) * (1.0 / l)) * (math.sin(k_m) * math.tan(k_m)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(1.0 / Float64(k_m / t_m))
	tmp = 0.0
	if ((t_m <= 9.6e-69) || !(t_m <= 1.25e+99))
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(l * Float64(sqrt(2.0) / (k_m ^ 2.0))) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 * Float64(t_2 * t_2)) / Float64(Float64(Float64((t_m ^ 3.0) / l) * Float64(1.0 / l)) * Float64(sin(k_m) * tan(k_m))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = 1.0 / (k_m / t_m);
	tmp = 0.0;
	if ((t_m <= 9.6e-69) || ~((t_m <= 1.25e+99)))
		tmp = (cos(k_m) / t_m) * ((l * (sqrt(2.0) / (k_m ^ 2.0))) ^ 2.0);
	else
		tmp = (2.0 * (t_2 * t_2)) / ((((t_m ^ 3.0) / l) * (1.0 / l)) * (sin(k_m) * tan(k_m)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(1.0 / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[Or[LessEqual[t$95$m, 9.6e-69], N[Not[LessEqual[t$95$m, 1.25e+99]], $MachinePrecision]], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.6000000000000005e-69 or 1.25000000000000002e99 < t

   1. Initial program 27.9%

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

   3. Simplified 36.1%

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

      1. add-sqr-sqrt 24.7%

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

      2. pow2 24.7%

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

   6. Applied egg-rr 20.9%

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

      1. *-commutative 20.9%

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

   8. Simplified 20.9%

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

   9. Taylor expanded in l around 0 51.4%

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

       1. times-frac 51.9%

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

   11. Simplified 51.9%

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

       1. unpow-prod-down 48.4%

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

       2. frac-times 48.4%

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

       3. pow2 48.4%

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

       4. add-sqr-sqrt 93.8%

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

   13. Applied egg-rr 93.8%

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

   14. Taylor expanded in k around 0 73.3%

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

       1. associate-/l* 73.3%

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

   16. Simplified 73.3%

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

   if 9.6000000000000005e-69 < t < 1.25000000000000002e99

   1. Initial program 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-/r* 56.5%

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

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv 60.8%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. +-rgt-identity 60.8%

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

      3. pow-flip 60.9%

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

      4. metadata-eval 60.9%

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

   6. Applied egg-rr 60.9%

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

      1. metadata-eval 60.9%

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

      2. pow-prod-up 60.8%

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

      3. unpow-1 60.8%

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

      4. unpow-1 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. associate-/r* 72.5%

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

      2. div-inv 72.5%

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

   10. Applied egg-rr 72.5%

       \[\leadsto \frac{2 \cdot \left(\frac{1}{\frac{k}{t}} \cdot \frac{1}{\frac{k}{t}}\right)}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{1}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.6 \cdot 10^{-69} \lor \neg \left(t \leq 1.25 \cdot 10^{+99}\right):\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{1}{\frac{k}{t}} \cdot \frac{1}{\frac{k}{t}}\right)}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{1}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1}{\frac{k\_m}{t\_m}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-68}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2.55 \cdot 10^{+98}:\\ \;\;\;\;\frac{2 \cdot \left(t\_2 \cdot t\_2\right)}{\left(\frac{{t\_m}^{3}}{\ell} \cdot \frac{1}{\ell}\right) \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t\_m}} \cdot \left(\ell \cdot \frac{\sqrt{2}}{{k\_m}^{2}}\right)\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ 1.0 (/ k_m t_m))))
   (*
    t_s
    (if (<= t_m 1.05e-68)
      (* (/ (cos k_m) t_m) (pow (/ (* l (sqrt 2.0)) (pow k_m 2.0)) 2.0))
      (if (<= t_m 2.55e+98)
        (/
         (* 2.0 (* t_2 t_2))
         (* (* (/ (pow t_m 3.0) l) (/ 1.0 l)) (* (sin k_m) (tan k_m))))
        (pow
         (* (sqrt (/ 1.0 t_m)) (* l (/ (sqrt 2.0) (pow k_m 2.0))))
         2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = 1.0 / (k_m / t_m);
	double tmp;
	if (t_m <= 1.05e-68) {
		tmp = (cos(k_m) / t_m) * pow(((l * sqrt(2.0)) / pow(k_m, 2.0)), 2.0);
	} else if (t_m <= 2.55e+98) {
		tmp = (2.0 * (t_2 * t_2)) / (((pow(t_m, 3.0) / l) * (1.0 / l)) * (sin(k_m) * tan(k_m)));
	} else {
		tmp = pow((sqrt((1.0 / t_m)) * (l * (sqrt(2.0) / pow(k_m, 2.0)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 / (k_m / t_m)
    if (t_m <= 1.05d-68) then
        tmp = (cos(k_m) / t_m) * (((l * sqrt(2.0d0)) / (k_m ** 2.0d0)) ** 2.0d0)
    else if (t_m <= 2.55d+98) then
        tmp = (2.0d0 * (t_2 * t_2)) / ((((t_m ** 3.0d0) / l) * (1.0d0 / l)) * (sin(k_m) * tan(k_m)))
    else
        tmp = (sqrt((1.0d0 / t_m)) * (l * (sqrt(2.0d0) / (k_m ** 2.0d0)))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = 1.0 / (k_m / t_m);
	double tmp;
	if (t_m <= 1.05e-68) {
		tmp = (Math.cos(k_m) / t_m) * Math.pow(((l * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)), 2.0);
	} else if (t_m <= 2.55e+98) {
		tmp = (2.0 * (t_2 * t_2)) / (((Math.pow(t_m, 3.0) / l) * (1.0 / l)) * (Math.sin(k_m) * Math.tan(k_m)));
	} else {
		tmp = Math.pow((Math.sqrt((1.0 / t_m)) * (l * (Math.sqrt(2.0) / Math.pow(k_m, 2.0)))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = 1.0 / (k_m / t_m)
	tmp = 0
	if t_m <= 1.05e-68:
		tmp = (math.cos(k_m) / t_m) * math.pow(((l * math.sqrt(2.0)) / math.pow(k_m, 2.0)), 2.0)
	elif t_m <= 2.55e+98:
		tmp = (2.0 * (t_2 * t_2)) / (((math.pow(t_m, 3.0) / l) * (1.0 / l)) * (math.sin(k_m) * math.tan(k_m)))
	else:
		tmp = math.pow((math.sqrt((1.0 / t_m)) * (l * (math.sqrt(2.0) / math.pow(k_m, 2.0)))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(1.0 / Float64(k_m / t_m))
	tmp = 0.0
	if (t_m <= 1.05e-68)
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(Float64(l * sqrt(2.0)) / (k_m ^ 2.0)) ^ 2.0));
	elseif (t_m <= 2.55e+98)
		tmp = Float64(Float64(2.0 * Float64(t_2 * t_2)) / Float64(Float64(Float64((t_m ^ 3.0) / l) * Float64(1.0 / l)) * Float64(sin(k_m) * tan(k_m))));
	else
		tmp = Float64(sqrt(Float64(1.0 / t_m)) * Float64(l * Float64(sqrt(2.0) / (k_m ^ 2.0)))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = 1.0 / (k_m / t_m);
	tmp = 0.0;
	if (t_m <= 1.05e-68)
		tmp = (cos(k_m) / t_m) * (((l * sqrt(2.0)) / (k_m ^ 2.0)) ^ 2.0);
	elseif (t_m <= 2.55e+98)
		tmp = (2.0 * (t_2 * t_2)) / ((((t_m ^ 3.0) / l) * (1.0 / l)) * (sin(k_m) * tan(k_m)));
	else
		tmp = (sqrt((1.0 / t_m)) * (l * (sqrt(2.0) / (k_m ^ 2.0)))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(1.0 / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.05e-68], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.55e+98], N[(N[(2.0 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.05000000000000004e-68

   1. Initial program 33.4%

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

   3. Simplified 37.6%

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

      1. add-sqr-sqrt 22.7%

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

      2. pow2 22.7%

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

   6. Applied egg-rr 18.2%

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

      1. *-commutative 18.2%

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

   8. Simplified 18.2%

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

   9. Taylor expanded in l around 0 43.8%

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

       1. times-frac 43.9%

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

   11. Simplified 43.9%

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

       1. unpow-prod-down 40.9%

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

       2. frac-times 40.9%

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

       3. pow2 40.9%

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

       4. add-sqr-sqrt 94.2%

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

   13. Applied egg-rr 94.2%

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

   14. Taylor expanded in k around 0 71.6%

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

   if 1.05000000000000004e-68 < t < 2.54999999999999994e98

   1. Initial program 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-/r* 56.5%

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

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv 60.8%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. +-rgt-identity 60.8%

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

      3. pow-flip 60.9%

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

      4. metadata-eval 60.9%

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

   6. Applied egg-rr 60.9%

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

      1. metadata-eval 60.9%

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

      2. pow-prod-up 60.8%

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

      3. unpow-1 60.8%

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

      4. unpow-1 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. associate-/r* 72.5%

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

      2. div-inv 72.5%

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

   10. Applied egg-rr 72.5%

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

   if 2.54999999999999994e98 < t

   1. Initial program 9.8%

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

   3. Simplified 31.1%

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

      1. add-sqr-sqrt 31.1%

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

      2. pow2 31.1%

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

   6. Applied egg-rr 29.6%

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

      1. *-commutative 29.6%

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

   8. Simplified 29.6%

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

   9. Taylor expanded in k around 0 80.4%

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

       1. associate-/l* 80.5%

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

   11. Simplified 80.5%

       \[\leadsto {\color{blue}{\left(\left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right) \cdot \sqrt{\frac{1}{t}}\right)}}^{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-68}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{+98}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{1}{\frac{k}{t}} \cdot \frac{1}{\frac{k}{t}}\right)}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{1}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{1}{t}} \cdot \left(\ell \cdot \frac{\sqrt{2}}{{k}^{2}}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1}{\frac{k\_m}{t\_m}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\cos k\_m}{t\_m} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{{k\_m}^{2}}\right)}^{2}\\ \mathbf{elif}\;t\_m \leq 2.4 \cdot 10^{+97}:\\ \;\;\;\;\frac{2 \cdot \left(t\_2 \cdot t\_2\right)}{\left(\frac{{t\_m}^{3}}{\ell} \cdot \frac{1}{\ell}\right) \cdot \left(\sin k\_m \cdot \tan k\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\frac{1}{t\_m}} \cdot \frac{\sqrt{2}}{{k\_m}^{2}}\right)\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ 1.0 (/ k_m t_m))))
   (*
    t_s
    (if (<= t_m 9.6e-69)
      (* (/ (cos k_m) t_m) (pow (/ (* l (sqrt 2.0)) (pow k_m 2.0)) 2.0))
      (if (<= t_m 2.4e+97)
        (/
         (* 2.0 (* t_2 t_2))
         (* (* (/ (pow t_m 3.0) l) (/ 1.0 l)) (* (sin k_m) (tan k_m))))
        (pow
         (* l (* (sqrt (/ 1.0 t_m)) (/ (sqrt 2.0) (pow k_m 2.0))))
         2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = 1.0 / (k_m / t_m);
	double tmp;
	if (t_m <= 9.6e-69) {
		tmp = (cos(k_m) / t_m) * pow(((l * sqrt(2.0)) / pow(k_m, 2.0)), 2.0);
	} else if (t_m <= 2.4e+97) {
		tmp = (2.0 * (t_2 * t_2)) / (((pow(t_m, 3.0) / l) * (1.0 / l)) * (sin(k_m) * tan(k_m)));
	} else {
		tmp = pow((l * (sqrt((1.0 / t_m)) * (sqrt(2.0) / pow(k_m, 2.0)))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 / (k_m / t_m)
    if (t_m <= 9.6d-69) then
        tmp = (cos(k_m) / t_m) * (((l * sqrt(2.0d0)) / (k_m ** 2.0d0)) ** 2.0d0)
    else if (t_m <= 2.4d+97) then
        tmp = (2.0d0 * (t_2 * t_2)) / ((((t_m ** 3.0d0) / l) * (1.0d0 / l)) * (sin(k_m) * tan(k_m)))
    else
        tmp = (l * (sqrt((1.0d0 / t_m)) * (sqrt(2.0d0) / (k_m ** 2.0d0)))) ** 2.0d0
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = 1.0 / (k_m / t_m);
	double tmp;
	if (t_m <= 9.6e-69) {
		tmp = (Math.cos(k_m) / t_m) * Math.pow(((l * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)), 2.0);
	} else if (t_m <= 2.4e+97) {
		tmp = (2.0 * (t_2 * t_2)) / (((Math.pow(t_m, 3.0) / l) * (1.0 / l)) * (Math.sin(k_m) * Math.tan(k_m)));
	} else {
		tmp = Math.pow((l * (Math.sqrt((1.0 / t_m)) * (Math.sqrt(2.0) / Math.pow(k_m, 2.0)))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = 1.0 / (k_m / t_m)
	tmp = 0
	if t_m <= 9.6e-69:
		tmp = (math.cos(k_m) / t_m) * math.pow(((l * math.sqrt(2.0)) / math.pow(k_m, 2.0)), 2.0)
	elif t_m <= 2.4e+97:
		tmp = (2.0 * (t_2 * t_2)) / (((math.pow(t_m, 3.0) / l) * (1.0 / l)) * (math.sin(k_m) * math.tan(k_m)))
	else:
		tmp = math.pow((l * (math.sqrt((1.0 / t_m)) * (math.sqrt(2.0) / math.pow(k_m, 2.0)))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(1.0 / Float64(k_m / t_m))
	tmp = 0.0
	if (t_m <= 9.6e-69)
		tmp = Float64(Float64(cos(k_m) / t_m) * (Float64(Float64(l * sqrt(2.0)) / (k_m ^ 2.0)) ^ 2.0));
	elseif (t_m <= 2.4e+97)
		tmp = Float64(Float64(2.0 * Float64(t_2 * t_2)) / Float64(Float64(Float64((t_m ^ 3.0) / l) * Float64(1.0 / l)) * Float64(sin(k_m) * tan(k_m))));
	else
		tmp = Float64(l * Float64(sqrt(Float64(1.0 / t_m)) * Float64(sqrt(2.0) / (k_m ^ 2.0)))) ^ 2.0;
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = 1.0 / (k_m / t_m);
	tmp = 0.0;
	if (t_m <= 9.6e-69)
		tmp = (cos(k_m) / t_m) * (((l * sqrt(2.0)) / (k_m ^ 2.0)) ^ 2.0);
	elseif (t_m <= 2.4e+97)
		tmp = (2.0 * (t_2 * t_2)) / ((((t_m ^ 3.0) / l) * (1.0 / l)) * (sin(k_m) * tan(k_m)));
	else
		tmp = (l * (sqrt((1.0 / t_m)) * (sqrt(2.0) / (k_m ^ 2.0)))) ^ 2.0;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(1.0 / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 9.6e-69], N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.4e+97], N[(N[(2.0 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(l * N[(N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 9.6000000000000005e-69

   1. Initial program 33.4%

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

   3. Simplified 37.6%

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

      1. add-sqr-sqrt 22.7%

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

      2. pow2 22.7%

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

   6. Applied egg-rr 18.2%

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

      1. *-commutative 18.2%

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

   8. Simplified 18.2%

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

   9. Taylor expanded in l around 0 43.8%

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

       1. times-frac 43.9%

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

   11. Simplified 43.9%

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

       1. unpow-prod-down 40.9%

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

       2. frac-times 40.9%

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

       3. pow2 40.9%

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

       4. add-sqr-sqrt 94.2%

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

   13. Applied egg-rr 94.2%

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

   14. Taylor expanded in k around 0 71.6%

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

   if 9.6000000000000005e-69 < t < 2.4e97

   1. Initial program 56.5%

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

      1. *-commutative 56.5%

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

      2. associate-/r* 56.5%

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

   3. Simplified 60.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv 60.8%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. +-rgt-identity 60.8%

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

      3. pow-flip 60.9%

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

      4. metadata-eval 60.9%

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

   6. Applied egg-rr 60.9%

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

      1. metadata-eval 60.9%

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

      2. pow-prod-up 60.8%

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

      3. unpow-1 60.8%

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

      4. unpow-1 60.8%

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

   8. Applied egg-rr 60.8%

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

      1. associate-/r* 72.5%

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

      2. div-inv 72.5%

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

   10. Applied egg-rr 72.5%

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

   if 2.4e97 < t

   1. Initial program 9.8%

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

   3. Simplified 31.1%

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

      1. add-sqr-sqrt 31.1%

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

      2. pow2 31.1%

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

   6. Applied egg-rr 29.6%

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

      1. *-commutative 29.6%

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

   8. Simplified 29.6%

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

   9. Taylor expanded in k around 0 80.5%

      \[\leadsto {\left(\ell \cdot \color{blue}{\left(\frac{\sqrt{2}}{{k}^{2}} \cdot \sqrt{\frac{1}{t}}\right)}\right)}^{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{{k}^{2}}\right)}^{2}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+97}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{1}{\frac{k}{t}} \cdot \frac{1}{\frac{k}{t}}\right)}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{1}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\ell \cdot \left(\sqrt{\frac{1}{t}} \cdot \frac{\sqrt{2}}{{k}^{2}}\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\cos k\_m}{t\_m}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k\_m \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k\_m} \cdot \frac{\sqrt{2}}{k\_m}\right) \cdot \sqrt{t\_2}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot {\left(\frac{\ell \cdot \sqrt{2}}{k\_m \cdot \sin k\_m}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ (cos k_m) t_m)))
   (*
    t_s
    (if (<= k_m 3.9e-19)
      (pow (* (* (/ l k_m) (/ (sqrt 2.0) k_m)) (sqrt t_2)) 2.0)
      (* t_2 (pow (/ (* l (sqrt 2.0)) (* k_m (sin k_m))) 2.0))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = cos(k_m) / t_m;
	double tmp;
	if (k_m <= 3.9e-19) {
		tmp = pow((((l / k_m) * (sqrt(2.0) / k_m)) * sqrt(t_2)), 2.0);
	} else {
		tmp = t_2 * pow(((l * sqrt(2.0)) / (k_m * sin(k_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = cos(k_m) / t_m
    if (k_m <= 3.9d-19) then
        tmp = (((l / k_m) * (sqrt(2.0d0) / k_m)) * sqrt(t_2)) ** 2.0d0
    else
        tmp = t_2 * (((l * sqrt(2.0d0)) / (k_m * sin(k_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = Math.cos(k_m) / t_m;
	double tmp;
	if (k_m <= 3.9e-19) {
		tmp = Math.pow((((l / k_m) * (Math.sqrt(2.0) / k_m)) * Math.sqrt(t_2)), 2.0);
	} else {
		tmp = t_2 * Math.pow(((l * Math.sqrt(2.0)) / (k_m * Math.sin(k_m))), 2.0);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = math.cos(k_m) / t_m
	tmp = 0
	if k_m <= 3.9e-19:
		tmp = math.pow((((l / k_m) * (math.sqrt(2.0) / k_m)) * math.sqrt(t_2)), 2.0)
	else:
		tmp = t_2 * math.pow(((l * math.sqrt(2.0)) / (k_m * math.sin(k_m))), 2.0)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(cos(k_m) / t_m)
	tmp = 0.0
	if (k_m <= 3.9e-19)
		tmp = Float64(Float64(Float64(l / k_m) * Float64(sqrt(2.0) / k_m)) * sqrt(t_2)) ^ 2.0;
	else
		tmp = Float64(t_2 * (Float64(Float64(l * sqrt(2.0)) / Float64(k_m * sin(k_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = cos(k_m) / t_m;
	tmp = 0.0;
	if (k_m <= 3.9e-19)
		tmp = (((l / k_m) * (sqrt(2.0) / k_m)) * sqrt(t_2)) ^ 2.0;
	else
		tmp = t_2 * (((l * sqrt(2.0)) / (k_m * sin(k_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[k$95$m, 3.9e-19], N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / k$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(t$95$2 * N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.89999999999999995e-19

   1. Initial program 33.3%

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

   3. Simplified 39.4%

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

      1. add-sqr-sqrt 26.6%

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

      2. pow2 26.6%

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

   6. Applied egg-rr 31.1%

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

      1. *-commutative 31.1%

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

   8. Simplified 31.1%

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

   9. Taylor expanded in l around 0 54.7%

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

       1. times-frac 55.9%

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

   11. Simplified 55.9%

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

   12. Taylor expanded in k around 0 47.3%

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

   if 3.89999999999999995e-19 < k

   1. Initial program 26.7%

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

   3. Simplified 37.2%

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

      1. add-sqr-sqrt 34.2%

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

      2. pow2 34.2%

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

   6. Applied egg-rr 19.3%

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

      1. *-commutative 19.3%

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

   8. Simplified 19.3%

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

   9. Taylor expanded in l around 0 55.7%

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

       1. times-frac 55.8%

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

   11. Simplified 55.8%

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

       1. unpow-prod-down 50.6%

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

       2. frac-times 50.7%

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

       3. pow2 50.7%

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

       4. add-sqr-sqrt 91.6%

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

   13. Applied egg-rr 91.6%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;{\left(\left(\frac{\ell}{k} \cdot \frac{\sqrt{2}}{k}\right) \cdot \sqrt{\frac{\cos k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{k \cdot \sin k}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.0% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (* (/ (cos k_m) t_m) (pow (/ (* (/ l k_m) (sqrt 2.0)) (sin k_m)) 2.0))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((Math.cos(k_m) / t_m) * Math.pow((((l / k_m) * Math.sqrt(2.0)) / Math.sin(k_m)), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(N[(l / k$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

   1. add-sqr-sqrt 28.7%

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

   2. pow2 28.7%

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

6. Applied egg-rr 27.8%

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

   1. *-commutative 27.8%

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

8. Simplified 27.8%

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

9. Taylor expanded in l around 0 55.0%

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

    1. times-frac 55.9%

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

11. Simplified 55.9%

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

    1. unpow-prod-down 52.8%

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

    2. frac-times 52.3%

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

    3. pow2 52.3%

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

    4. add-sqr-sqrt 93.5%

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

13. Applied egg-rr 93.5%

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

14. Taylor expanded in l around 0 68.2%

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

    1. times-frac 69.3%

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

    2. unpow2 69.3%

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

    3. rem-square-sqrt 69.4%

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

    4. rem-square-sqrt 69.3%

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

    5. unpow2 69.3%

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

    6. times-frac 69.3%

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

    7. unpow2 69.3%

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

    8. unpow2 69.3%

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

    9. times-frac 92.6%

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

    10. unswap-sqr 94.0%

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

    11. times-frac 93.6%

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

    12. associate-*r/ 93.3%

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

    13. times-frac 93.3%

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

    14. associate-*r/ 93.3%

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

16. Simplified 94.0%

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

17. Final simplification 94.0%

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

Alternative 7: 93.0% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (* (/ (cos k_m) t_m) (pow (* (/ l k_m) (/ (sqrt 2.0) (sin k_m))) 2.0))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((Math.cos(k_m) / t_m) * Math.pow(((l / k_m) * (Math.sqrt(2.0) / Math.sin(k_m))), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l / k$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

   1. add-sqr-sqrt 28.7%

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

   2. pow2 28.7%

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

6. Applied egg-rr 27.8%

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

   1. *-commutative 27.8%

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

8. Simplified 27.8%

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

9. Taylor expanded in l around 0 55.0%

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

    1. times-frac 55.9%

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

11. Simplified 55.9%

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

    1. unpow-prod-down 52.8%

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

    2. frac-times 52.3%

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

    3. pow2 52.3%

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

    4. add-sqr-sqrt 93.5%

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

13. Applied egg-rr 93.5%

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

    1. times-frac 94.0%

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

15. Applied egg-rr 94.0%

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

16. Final simplification 94.0%

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

Alternative 8: 92.1% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (* (/ (cos k_m) t_m) (pow (* (sqrt 2.0) (/ l (* k_m (sin k_m)))) 2.0))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((Math.cos(k_m) / t_m) * Math.pow((Math.sqrt(2.0) * (l / (k_m * Math.sin(k_m)))), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[(l / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

   1. add-sqr-sqrt 28.7%

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

   2. pow2 28.7%

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

6. Applied egg-rr 27.8%

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

   1. *-commutative 27.8%

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

8. Simplified 27.8%

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

9. Taylor expanded in l around 0 55.0%

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

    1. times-frac 55.9%

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

11. Simplified 55.9%

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

    1. *-commutative 55.9%

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

    2. unpow-prod-down 52.8%

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

    3. pow2 52.8%

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

    4. add-sqr-sqrt 94.0%

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

    5. frac-times 93.5%

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

13. Applied egg-rr 93.5%

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

    1. *-commutative 93.5%

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

    2. associate-/l* 93.5%

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

15. Simplified 93.5%

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

Alternative 9: 92.1% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (* (/ (cos k_m) t_m) (pow (* l (/ (sqrt 2.0) (* k_m (sin k_m)))) 2.0))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((Math.cos(k_m) / t_m) * Math.pow((l * (Math.sqrt(2.0) / (k_m * Math.sin(k_m)))), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(l * N[(N[Sqrt[2.0], $MachinePrecision] / N[(k$95$m * N[Sin[k$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

   1. add-sqr-sqrt 28.7%

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

   2. pow2 28.7%

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

6. Applied egg-rr 27.8%

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

   1. *-commutative 27.8%

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

8. Simplified 27.8%

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

9. Taylor expanded in l around 0 55.0%

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

    1. times-frac 55.9%

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

11. Simplified 55.9%

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

    1. *-commutative 55.9%

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

    2. unpow-prod-down 52.8%

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

    3. pow2 52.8%

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

    4. add-sqr-sqrt 94.0%

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

    5. frac-times 93.5%

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

13. Applied egg-rr 93.5%

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

    1. associate-/l* 93.3%

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

15. Simplified 93.3%

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

Alternative 10: 74.9% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (pow (* (/ (* l (sqrt 2.0)) (pow k_m 2.0)) (sqrt (/ 1.0 t_m))) 2.0)))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * Math.pow((((l * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)) * Math.sqrt((1.0 / t_m))), 2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[Power[N[(N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

   1. add-sqr-sqrt 28.7%

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

   2. pow2 28.7%

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

6. Applied egg-rr 27.8%

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

   1. *-commutative 27.8%

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

8. Simplified 27.8%

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

9. Taylor expanded in k around 0 38.2%

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

Alternative 11: 74.7% accurate, 1.0× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (/ (cos k_m) t_m) (pow (/ (* l (sqrt 2.0)) (pow k_m 2.0)) 2.0))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((Math.cos(k_m) / t_m) * Math.pow(((l * Math.sqrt(2.0)) / Math.pow(k_m, 2.0)), 2.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(N[Cos[k$95$m], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Power[N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

   1. add-sqr-sqrt 28.7%

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

   2. pow2 28.7%

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

6. Applied egg-rr 27.8%

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

   1. *-commutative 27.8%

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

8. Simplified 27.8%

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

9. Taylor expanded in l around 0 55.0%

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

    1. times-frac 55.9%

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

11. Simplified 55.9%

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

    1. unpow-prod-down 52.8%

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

    2. frac-times 52.3%

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

    3. pow2 52.3%

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

    4. add-sqr-sqrt 93.5%

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

13. Applied egg-rr 93.5%

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

14. Taylor expanded in k around 0 71.7%

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

15. Final simplification 71.7%

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

Alternative 12: 69.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{1}{\frac{k\_m}{t\_m}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \left({k\_m}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t\_m \leq 6 \cdot 10^{+98}:\\ \;\;\;\;\frac{2 \cdot \left(t\_2 \cdot t\_2\right)}{\left(\sin k\_m \cdot \tan k\_m\right) \cdot \left(\frac{{t\_m}^{2}}{\ell} \cdot \frac{t\_m}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}}}\\ \end{array} \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (let* ((t_2 (/ 1.0 (/ k_m t_m))))
   (*
    t_s
    (if (<= t_m 4e-154)
      (/ 2.0 (* (pow k_m 2.0) (* (pow k_m 2.0) (/ t_m (pow l 2.0)))))
      (if (<= t_m 6e+98)
        (/
         (* 2.0 (* t_2 t_2))
         (* (* (sin k_m) (tan k_m)) (* (/ (pow t_m 2.0) l) (/ t_m l))))
        (/ 2.0 (* (pow k_m 2.0) (/ (* t_m (pow k_m 2.0)) (pow l 2.0)))))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double t_2 = 1.0 / (k_m / t_m);
	double tmp;
	if (t_m <= 4e-154) {
		tmp = 2.0 / (pow(k_m, 2.0) * (pow(k_m, 2.0) * (t_m / pow(l, 2.0))));
	} else if (t_m <= 6e+98) {
		tmp = (2.0 * (t_2 * t_2)) / ((sin(k_m) * tan(k_m)) * ((pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) * ((t_m * pow(k_m, 2.0)) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = 1.0d0 / (k_m / t_m)
    if (t_m <= 4d-154) then
        tmp = 2.0d0 / ((k_m ** 2.0d0) * ((k_m ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else if (t_m <= 6d+98) then
        tmp = (2.0d0 * (t_2 * t_2)) / ((sin(k_m) * tan(k_m)) * (((t_m ** 2.0d0) / l) * (t_m / l)))
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) * ((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double t_2 = 1.0 / (k_m / t_m);
	double tmp;
	if (t_m <= 4e-154) {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * (Math.pow(k_m, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else if (t_m <= 6e+98) {
		tmp = (2.0 * (t_2 * t_2)) / ((Math.sin(k_m) * Math.tan(k_m)) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)));
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * ((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	t_2 = 1.0 / (k_m / t_m)
	tmp = 0
	if t_m <= 4e-154:
		tmp = 2.0 / (math.pow(k_m, 2.0) * (math.pow(k_m, 2.0) * (t_m / math.pow(l, 2.0))))
	elif t_m <= 6e+98:
		tmp = (2.0 * (t_2 * t_2)) / ((math.sin(k_m) * math.tan(k_m)) * ((math.pow(t_m, 2.0) / l) * (t_m / l)))
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) * ((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	t_2 = Float64(1.0 / Float64(k_m / t_m))
	tmp = 0.0
	if (t_m <= 4e-154)
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64((k_m ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	elseif (t_m <= 6e+98)
		tmp = Float64(Float64(2.0 * Float64(t_2 * t_2)) / Float64(Float64(sin(k_m) * tan(k_m)) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	t_2 = 1.0 / (k_m / t_m);
	tmp = 0.0;
	if (t_m <= 4e-154)
		tmp = 2.0 / ((k_m ^ 2.0) * ((k_m ^ 2.0) * (t_m / (l ^ 2.0))));
	elseif (t_m <= 6e+98)
		tmp = (2.0 * (t_2 * t_2)) / ((sin(k_m) * tan(k_m)) * (((t_m ^ 2.0) / l) * (t_m / l)));
	else
		tmp = 2.0 / ((k_m ^ 2.0) * ((t_m * (k_m ^ 2.0)) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := Block[{t$95$2 = N[(1.0 / N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 4e-154], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 6e+98], N[(N[(2.0 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k$95$m], $MachinePrecision] * N[Tan[k$95$m], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[t$95$m, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 3.9999999999999999e-154

   1. Initial program 32.7%

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

   3. Taylor expanded in t around 0 68.4%

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

      1. associate-/l* 70.9%

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

      2. *-commutative 70.9%

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

      3. times-frac 70.6%

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

   5. Simplified 70.6%

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

   6. Taylor expanded in k around 0 61.1%

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

      1. associate-/l* 59.8%

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

   8. Simplified 59.8%

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

   if 3.9999999999999999e-154 < t < 6.0000000000000003e98

   1. Initial program 49.2%

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

      1. *-commutative 49.2%

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

      2. associate-/r* 49.2%

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

   3. Simplified 51.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. div-inv 51.9%

         \[\leadsto \frac{\color{blue}{2 \cdot \frac{1}{{\left(\frac{k}{t}\right)}^{2} + 0}}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)} \]

      2. +-rgt-identity 51.9%

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

      3. pow-flip 51.9%

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

      4. metadata-eval 51.9%

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

   6. Applied egg-rr 51.9%

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

      1. metadata-eval 51.9%

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

      2. pow-prod-up 51.9%

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

      3. unpow-1 51.9%

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

      4. unpow-1 51.9%

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

   8. Applied egg-rr 51.9%

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

      1. unpow3 51.9%

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

      2. times-frac 66.4%

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

      3. pow2 66.4%

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

   10. Applied egg-rr 66.4%

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

   if 6.0000000000000003e98 < t

   1. Initial program 9.8%

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

   3. Taylor expanded in t around 0 79.2%

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

      1. associate-/l* 79.4%

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

      2. *-commutative 79.4%

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

      3. times-frac 79.4%

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

   5. Simplified 79.4%

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

   6. Taylor expanded in k around 0 72.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+98}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{1}{\frac{k}{t}} \cdot \frac{1}{\frac{k}{t}}\right)}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{{t}^{2}}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 58.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.55 \cdot 10^{-185}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(k\_m \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \frac{1}{\frac{t\_m}{k\_m} \cdot \frac{t\_m}{k\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 2.55e-185)
    (/
     2.0
     (*
      (* (tan k_m) (* k_m (pow (/ (pow t_m 1.5) l) 2.0)))
      (/ 1.0 (* (/ t_m k_m) (/ t_m k_m)))))
    (/ 2.0 (* (pow k_m 2.0) (/ (* t_m (pow k_m 2.0)) (pow l 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 2.55e-185) {
		tmp = 2.0 / ((tan(k_m) * (k_m * pow((pow(t_m, 1.5) / l), 2.0))) * (1.0 / ((t_m / k_m) * (t_m / k_m))));
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) * ((t_m * pow(k_m, 2.0)) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 2.55d-185) then
        tmp = 2.0d0 / ((tan(k_m) * (k_m * (((t_m ** 1.5d0) / l) ** 2.0d0))) * (1.0d0 / ((t_m / k_m) * (t_m / k_m))))
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) * ((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (l <= 2.55e-185) {
		tmp = 2.0 / ((Math.tan(k_m) * (k_m * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * (1.0 / ((t_m / k_m) * (t_m / k_m))));
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * ((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 2.55e-185:
		tmp = 2.0 / ((math.tan(k_m) * (k_m * math.pow((math.pow(t_m, 1.5) / l), 2.0))) * (1.0 / ((t_m / k_m) * (t_m / k_m))))
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) * ((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 2.55e-185)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(k_m * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(1.0 / Float64(Float64(t_m / k_m) * Float64(t_m / k_m)))));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 2.55e-185)
		tmp = 2.0 / ((tan(k_m) * (k_m * (((t_m ^ 1.5) / l) ^ 2.0))) * (1.0 / ((t_m / k_m) * (t_m / k_m))));
	else
		tmp = 2.0 / ((k_m ^ 2.0) * ((t_m * (k_m ^ 2.0)) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[l, 2.55e-185], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(t$95$m / k$95$m), $MachinePrecision] * N[(t$95$m / k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.5500000000000002e-185

   1. Initial program 31.3%

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

   3. Taylor expanded in k around 0 30.9%

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

      1. associate-/l* 30.3%

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

      2. metadata-eval 30.3%

         \[\leadsto \frac{2}{\left(\left(k \cdot \frac{{t}^{\color{blue}{\left(2 \cdot 1.5\right)}}}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      3. pow-sqr 14.4%

         \[\leadsto \frac{2}{\left(\left(k \cdot \frac{\color{blue}{{t}^{1.5} \cdot {t}^{1.5}}}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      4. unpow2 14.4%

         \[\leadsto \frac{2}{\left(\left(k \cdot \frac{{t}^{1.5} \cdot {t}^{1.5}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. times-frac 20.7%

         \[\leadsto \frac{2}{\left(\left(k \cdot \color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      6. unpow2 20.7%

         \[\leadsto \frac{2}{\left(\left(k \cdot \color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. Simplified 20.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   6. Step-by-step derivation

      1. add-exp-log 20.6%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \left(\color{blue}{e^{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)} \]

      2. expm1-define 20.6%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      3. log1p-define 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

      4. unpow2 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{k}{t} \cdot \frac{k}{t}}\right)\right)} \]

      5. expm1-log1p-u 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}} \]

      6. clear-num 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \left(\color{blue}{\frac{1}{\frac{t}{k}}} \cdot \frac{k}{t}\right)} \]

      7. clear-num 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \left(\frac{1}{\frac{t}{k}} \cdot \color{blue}{\frac{1}{\frac{t}{k}}}\right)} \]

      8. frac-times 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \color{blue}{\frac{1 \cdot 1}{\frac{t}{k} \cdot \frac{t}{k}}}} \]

      9. metadata-eval 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \frac{\color{blue}{1}}{\frac{t}{k} \cdot \frac{t}{k}}} \]

   7. Applied egg-rr 26.1%

      \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \color{blue}{\frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}} \]

   if 2.5500000000000002e-185 < l

   1. Initial program 31.6%

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

   3. Taylor expanded in t around 0 72.2%

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

      1. associate-/l* 73.1%

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

      2. *-commutative 73.1%

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

      3. times-frac 73.1%

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

   5. Simplified 73.1%

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

   6. Taylor expanded in k around 0 56.9%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.55 \cdot 10^{-185}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 4.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{2}{\left(\tan k\_m \cdot \left(k\_m \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(\frac{k\_m}{t\_m} \cdot \frac{k\_m}{t\_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 4.6e-185)
    (/
     2.0
     (*
      (* (tan k_m) (* k_m (pow (/ (pow t_m 1.5) l) 2.0)))
      (* (/ k_m t_m) (/ k_m t_m))))
    (/ 2.0 (* (pow k_m 2.0) (/ (* t_m (pow k_m 2.0)) (pow l 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 4.6e-185) {
		tmp = 2.0 / ((tan(k_m) * (k_m * pow((pow(t_m, 1.5) / l), 2.0))) * ((k_m / t_m) * (k_m / t_m)));
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) * ((t_m * pow(k_m, 2.0)) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 4.6d-185) then
        tmp = 2.0d0 / ((tan(k_m) * (k_m * (((t_m ** 1.5d0) / l) ** 2.0d0))) * ((k_m / t_m) * (k_m / t_m)))
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) * ((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (l <= 4.6e-185) {
		tmp = 2.0 / ((Math.tan(k_m) * (k_m * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * ((k_m / t_m) * (k_m / t_m)));
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * ((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 4.6e-185:
		tmp = 2.0 / ((math.tan(k_m) * (k_m * math.pow((math.pow(t_m, 1.5) / l), 2.0))) * ((k_m / t_m) * (k_m / t_m)))
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) * ((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 4.6e-185)
		tmp = Float64(2.0 / Float64(Float64(tan(k_m) * Float64(k_m * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(Float64(k_m / t_m) * Float64(k_m / t_m))));
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 4.6e-185)
		tmp = 2.0 / ((tan(k_m) * (k_m * (((t_m ^ 1.5) / l) ^ 2.0))) * ((k_m / t_m) * (k_m / t_m)));
	else
		tmp = 2.0 / ((k_m ^ 2.0) * ((t_m * (k_m ^ 2.0)) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[l, 4.6e-185], N[(2.0 / N[(N[(N[Tan[k$95$m], $MachinePrecision] * N[(k$95$m * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k$95$m / t$95$m), $MachinePrecision] * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.6000000000000002e-185

   1. Initial program 31.3%

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

   3. Taylor expanded in k around 0 30.9%

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

      1. associate-/l* 30.3%

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

      2. metadata-eval 30.3%

         \[\leadsto \frac{2}{\left(\left(k \cdot \frac{{t}^{\color{blue}{\left(2 \cdot 1.5\right)}}}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      3. pow-sqr 14.4%

         \[\leadsto \frac{2}{\left(\left(k \cdot \frac{\color{blue}{{t}^{1.5} \cdot {t}^{1.5}}}{{\ell}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      4. unpow2 14.4%

         \[\leadsto \frac{2}{\left(\left(k \cdot \frac{{t}^{1.5} \cdot {t}^{1.5}}{\color{blue}{\ell \cdot \ell}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      5. times-frac 20.7%

         \[\leadsto \frac{2}{\left(\left(k \cdot \color{blue}{\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

      6. unpow2 20.7%

         \[\leadsto \frac{2}{\left(\left(k \cdot \color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]

   5. Simplified 20.7%

      \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   6. Step-by-step derivation

      1. add-exp-log 20.6%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \left(\color{blue}{e^{\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} - 1\right)} \]

      2. expm1-define 20.6%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \color{blue}{\mathsf{expm1}\left(\log \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

      3. log1p-define 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left({\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]

      4. expm1-log1p-u 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

      5. unpow2 26.1%

         \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}} \]

   7. Applied egg-rr 26.1%

      \[\leadsto \frac{2}{\left(\left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right) \cdot \color{blue}{\left(\frac{k}{t} \cdot \frac{k}{t}\right)}} \]

   if 4.6000000000000002e-185 < l

   1. Initial program 31.6%

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

   3. Taylor expanded in t around 0 72.2%

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

      1. associate-/l* 73.1%

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

      2. *-commutative 73.1%

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

      3. times-frac 73.1%

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

   5. Simplified 73.1%

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

   6. Taylor expanded in k around 0 56.9%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(\frac{k}{t} \cdot \frac{k}{t}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-170}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t\_m}^{3}}}}{k\_m \cdot \frac{k\_m}{t\_m}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \frac{t\_m \cdot {k\_m}^{2}}{{\ell}^{2}}}\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= l 2.5e-170)
    (pow (* l (/ (sqrt (/ 2.0 (pow t_m 3.0))) (* k_m (/ k_m t_m)))) 2.0)
    (/ 2.0 (* (pow k_m 2.0) (/ (* t_m (pow k_m 2.0)) (pow l 2.0)))))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (l <= 2.5e-170) {
		tmp = pow((l * (sqrt((2.0 / pow(t_m, 3.0))) / (k_m * (k_m / t_m)))), 2.0);
	} else {
		tmp = 2.0 / (pow(k_m, 2.0) * ((t_m * pow(k_m, 2.0)) / pow(l, 2.0)));
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (l <= 2.5d-170) then
        tmp = (l * (sqrt((2.0d0 / (t_m ** 3.0d0))) / (k_m * (k_m / t_m)))) ** 2.0d0
    else
        tmp = 2.0d0 / ((k_m ** 2.0d0) * ((t_m * (k_m ** 2.0d0)) / (l ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (l <= 2.5e-170) {
		tmp = Math.pow((l * (Math.sqrt((2.0 / Math.pow(t_m, 3.0))) / (k_m * (k_m / t_m)))), 2.0);
	} else {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * ((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0)));
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if l <= 2.5e-170:
		tmp = math.pow((l * (math.sqrt((2.0 / math.pow(t_m, 3.0))) / (k_m * (k_m / t_m)))), 2.0)
	else:
		tmp = 2.0 / (math.pow(k_m, 2.0) * ((t_m * math.pow(k_m, 2.0)) / math.pow(l, 2.0)))
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (l <= 2.5e-170)
		tmp = Float64(l * Float64(sqrt(Float64(2.0 / (t_m ^ 3.0))) / Float64(k_m * Float64(k_m / t_m)))) ^ 2.0;
	else
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64(Float64(t_m * (k_m ^ 2.0)) / (l ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (l <= 2.5e-170)
		tmp = (l * (sqrt((2.0 / (t_m ^ 3.0))) / (k_m * (k_m / t_m)))) ^ 2.0;
	else
		tmp = 2.0 / ((k_m ^ 2.0) * ((t_m * (k_m ^ 2.0)) / (l ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[l, 2.5e-170], N[Power[N[(l * N[(N[Sqrt[N[(2.0 / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(k$95$m * N[(k$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 2.50000000000000005e-170

   1. Initial program 30.3%

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

   3. Simplified 38.1%

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

      1. add-sqr-sqrt 27.5%

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

      2. pow2 27.5%

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

   6. Applied egg-rr 29.5%

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

      1. *-commutative 29.5%

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

   8. Simplified 29.5%

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

   9. Taylor expanded in k around 0 34.7%

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

   if 2.50000000000000005e-170 < l

   1. Initial program 33.2%

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

   3. Taylor expanded in t around 0 73.7%

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

      1. associate-/l* 74.6%

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

      2. *-commutative 74.6%

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

      3. times-frac 74.6%

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

   5. Simplified 74.6%

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

   6. Taylor expanded in k around 0 57.6%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-170}:\\ \;\;\;\;{\left(\ell \cdot \frac{\sqrt{\frac{2}{{t}^{3}}}}{k \cdot \frac{k}{t}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{{\ell}^{2}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{{k\_m}^{2} \cdot \left({k\_m}^{2} \cdot \frac{t\_m}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k\_m}^{4} \cdot \left(t\_m + {k\_m}^{2} \cdot \left(0.08611111111111111 \cdot \left(t\_m \cdot \left(k\_m \cdot k\_m\right)\right) + t\_m \cdot 0.16666666666666666\right)\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 9.2e+187)
    (/ 2.0 (* (pow k_m 2.0) (* (pow k_m 2.0) (/ t_m (pow l 2.0)))))
    (*
     (/
      2.0
      (*
       (pow k_m 4.0)
       (+
        t_m
        (*
         (pow k_m 2.0)
         (+
          (* 0.08611111111111111 (* t_m (* k_m k_m)))
          (* t_m 0.16666666666666666))))))
     (* l l)))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	double tmp;
	if (t_m <= 9.2e+187) {
		tmp = 2.0 / (pow(k_m, 2.0) * (pow(k_m, 2.0) * (t_m / pow(l, 2.0))));
	} else {
		tmp = (2.0 / (pow(k_m, 4.0) * (t_m + (pow(k_m, 2.0) * ((0.08611111111111111 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666)))))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    real(8) :: tmp
    if (t_m <= 9.2d+187) then
        tmp = 2.0d0 / ((k_m ** 2.0d0) * ((k_m ** 2.0d0) * (t_m / (l ** 2.0d0))))
    else
        tmp = (2.0d0 / ((k_m ** 4.0d0) * (t_m + ((k_m ** 2.0d0) * ((0.08611111111111111d0 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666d0)))))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

k_m = Math.abs(k);
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_m) {
	double tmp;
	if (t_m <= 9.2e+187) {
		tmp = 2.0 / (Math.pow(k_m, 2.0) * (Math.pow(k_m, 2.0) * (t_m / Math.pow(l, 2.0))));
	} else {
		tmp = (2.0 / (Math.pow(k_m, 4.0) * (t_m + (Math.pow(k_m, 2.0) * ((0.08611111111111111 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666)))))) * (l * l);
	}
	return t_s * tmp;
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	tmp = 0
	if t_m <= 9.2e+187:
		tmp = 2.0 / (math.pow(k_m, 2.0) * (math.pow(k_m, 2.0) * (t_m / math.pow(l, 2.0))))
	else:
		tmp = (2.0 / (math.pow(k_m, 4.0) * (t_m + (math.pow(k_m, 2.0) * ((0.08611111111111111 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666)))))) * (l * l)
	return t_s * tmp

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	tmp = 0.0
	if (t_m <= 9.2e+187)
		tmp = Float64(2.0 / Float64((k_m ^ 2.0) * Float64((k_m ^ 2.0) * Float64(t_m / (l ^ 2.0)))));
	else
		tmp = Float64(Float64(2.0 / Float64((k_m ^ 4.0) * Float64(t_m + Float64((k_m ^ 2.0) * Float64(Float64(0.08611111111111111 * Float64(t_m * Float64(k_m * k_m))) + Float64(t_m * 0.16666666666666666)))))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k_m)
	tmp = 0.0;
	if (t_m <= 9.2e+187)
		tmp = 2.0 / ((k_m ^ 2.0) * ((k_m ^ 2.0) * (t_m / (l ^ 2.0))));
	else
		tmp = (2.0 / ((k_m ^ 4.0) * (t_m + ((k_m ^ 2.0) * ((0.08611111111111111 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666)))))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 9.2e+187], N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(t$95$m / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[(t$95$m + N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(0.08611111111111111 * N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 9.20000000000000015e187

   1. Initial program 34.3%

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

   3. Taylor expanded in t around 0 66.4%

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

      1. associate-/l* 68.0%

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

      2. *-commutative 68.0%

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

      3. times-frac 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in k around 0 58.6%

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

      1. associate-/l* 57.7%

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

   8. Simplified 57.7%

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

   if 9.20000000000000015e187 < t

   1. Initial program 10.0%

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

   3. Simplified 33.3%

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

   5. Taylor expanded in k around 0 79.2%

      \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(0.08611111111111111 \cdot \left({k}^{2} \cdot t\right) + 0.16666666666666666 \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
   6. Step-by-step derivation

      1. unpow2 79.2%

         \[\leadsto \frac{2}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(0.08611111111111111 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) + 0.16666666666666666 \cdot t\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

   7. Applied egg-rr 79.2%

      \[\leadsto \frac{2}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(0.08611111111111111 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) + 0.16666666666666666 \cdot t\right)\right)} \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9.2 \cdot 10^{+187}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left({k}^{2} \cdot \frac{t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(0.08611111111111111 \cdot \left(t \cdot \left(k \cdot k\right)\right) + t \cdot 0.16666666666666666\right)\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 65.4% accurate, 1.3× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ 2.0 (* (pow k_m 2.0) (/ (* t_m (pow k_m 2.0)) (pow l 2.0))))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * (2.0 / (Math.pow(k_m, 2.0) * ((t_m * Math.pow(k_m, 2.0)) / Math.pow(l, 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(2.0 / N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(t$95$m * N[Power[k$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Taylor expanded in t around 0 68.7%

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

   1. associate-/l* 70.3%

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

   2. *-commutative 70.3%

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

   3. times-frac 70.1%

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

5. Simplified 70.1%

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

6. Taylor expanded in k around 0 60.9%

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

7. Final simplification 60.9%

   \[\leadsto \frac{2}{{k}^{2} \cdot \frac{t \cdot {k}^{2}}{{\ell}^{2}}} \]
8. Add Preprocessing

Alternative 18: 62.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{2}{{k\_m}^{4} \cdot \left(t\_m + {k\_m}^{2} \cdot \left(0.08611111111111111 \cdot \left(t\_m \cdot \left(k\_m \cdot k\_m\right)\right) + t\_m \cdot 0.16666666666666666\right)\right)} \cdot \left(\ell \cdot \ell\right)\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (*
  t_s
  (*
   (/
    2.0
    (*
     (pow k_m 4.0)
     (+
      t_m
      (*
       (pow k_m 2.0)
       (+
        (* 0.08611111111111111 (* t_m (* k_m k_m)))
        (* t_m 0.16666666666666666))))))
   (* l l))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((2.0 / (pow(k_m, 4.0) * (t_m + (pow(k_m, 2.0) * ((0.08611111111111111 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666)))))) * (l * l));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((2.0d0 / ((k_m ** 4.0d0) * (t_m + ((k_m ** 2.0d0) * ((0.08611111111111111d0 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666d0)))))) * (l * l))
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((2.0 / (Math.pow(k_m, 4.0) * (t_m + (Math.pow(k_m, 2.0) * ((0.08611111111111111 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666)))))) * (l * l));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((2.0 / (math.pow(k_m, 4.0) * (t_m + (math.pow(k_m, 2.0) * ((0.08611111111111111 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666)))))) * (l * l))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(2.0 / Float64((k_m ^ 4.0) * Float64(t_m + Float64((k_m ^ 2.0) * Float64(Float64(0.08611111111111111 * Float64(t_m * Float64(k_m * k_m))) + Float64(t_m * 0.16666666666666666)))))) * Float64(l * l)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((2.0 / ((k_m ^ 4.0) * (t_m + ((k_m ^ 2.0) * ((0.08611111111111111 * (t_m * (k_m * k_m))) + (t_m * 0.16666666666666666)))))) * (l * l));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(2.0 / N[(N[Power[k$95$m, 4.0], $MachinePrecision] * N[(t$95$m + N[(N[Power[k$95$m, 2.0], $MachinePrecision] * N[(N[(0.08611111111111111 * N[(t$95$m * N[(k$95$m * k$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$m * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

5. Taylor expanded in k around 0 58.6%

   \[\leadsto \frac{2}{\color{blue}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(0.08611111111111111 \cdot \left({k}^{2} \cdot t\right) + 0.16666666666666666 \cdot t\right)\right)}} \cdot \left(\ell \cdot \ell\right) \]
6. Step-by-step derivation

   1. unpow2 58.6%

      \[\leadsto \frac{2}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(0.08611111111111111 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) + 0.16666666666666666 \cdot t\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

7. Applied egg-rr 58.6%

   \[\leadsto \frac{2}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(0.08611111111111111 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) + 0.16666666666666666 \cdot t\right)\right)} \cdot \left(\ell \cdot \ell\right) \]

8. Final simplification 58.6%

   \[\leadsto \frac{2}{{k}^{4} \cdot \left(t + {k}^{2} \cdot \left(0.08611111111111111 \cdot \left(t \cdot \left(k \cdot k\right)\right) + t \cdot 0.16666666666666666\right)\right)} \cdot \left(\ell \cdot \ell\right) \]
9. Add Preprocessing

Alternative 19: 62.4% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ (/ 2.0 t_m) (pow k_m 4.0)))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((l * l) * ((2.0 / t_m) / Math.pow(k_m, 4.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] / N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

5. Taylor expanded in k around 0 58.5%

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

   1. add-cube-cbrt 58.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\sqrt[3]{{k}^{4} \cdot t} \cdot \sqrt[3]{{k}^{4} \cdot t}\right) \cdot \sqrt[3]{{k}^{4} \cdot t}}} \cdot \left(\ell \cdot \ell\right) \]

   2. pow3 58.5%

      \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{{k}^{4} \cdot t}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

   3. *-commutative 58.5%

      \[\leadsto \frac{2}{{\left(\sqrt[3]{\color{blue}{t \cdot {k}^{4}}}\right)}^{3}} \cdot \left(\ell \cdot \ell\right) \]

7. Applied egg-rr 58.5%

   \[\leadsto \frac{2}{\color{blue}{{\left(\sqrt[3]{t \cdot {k}^{4}}\right)}^{3}}} \cdot \left(\ell \cdot \ell\right) \]

8. Taylor expanded in t around 0 58.5%

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

   1. associate-/l/ 58.5%

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

10. Simplified 58.5%

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

11. Final simplification 58.5%

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

Alternative 20: 62.4% accurate, 3.8× speedup

Math

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

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ 2.0 (* t_m (pow k_m 4.0))))))

C

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

Fortran

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

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((l * l) * (2.0 / (t_m * Math.pow(k_m, 4.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(t$95$m * N[Power[k$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

5. Taylor expanded in k around 0 58.5%

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

6. Final simplification 58.5%

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

Alternative 21: 20.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \frac{{\ell}^{2} \cdot -0.11666666666666667}{t\_m} \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (/ (* (pow l 2.0) -0.11666666666666667) t_m)))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((pow(l, 2.0) * -0.11666666666666667) / t_m);
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * (((l ** 2.0d0) * (-0.11666666666666667d0)) / t_m)
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((Math.pow(l, 2.0) * -0.11666666666666667) / t_m);
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((math.pow(l, 2.0) * -0.11666666666666667) / t_m)

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64((l ^ 2.0) * -0.11666666666666667) / t_m))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * (((l ^ 2.0) * -0.11666666666666667) / t_m);
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(N[Power[l, 2.0], $MachinePrecision] * -0.11666666666666667), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

5. Taylor expanded in k around 0 36.3%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 19.8%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]
7. Step-by-step derivation

   1. associate-*l/ 19.8%

      \[\leadsto \color{blue}{\frac{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}{t}} \]

   2. pow2 19.8%

      \[\leadsto \frac{-0.11666666666666667 \cdot \color{blue}{{\ell}^{2}}}{t} \]

8. Applied egg-rr 19.8%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667 \cdot {\ell}^{2}}{t}} \]

9. Final simplification 19.8%

   \[\leadsto \frac{{\ell}^{2} \cdot -0.11666666666666667}{t} \]
10. Add Preprocessing

Alternative 22: 20.5% accurate, 60.1× speedup

Math

\[\begin{array}{l} k_m = \left|k\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t\_m}\right) \end{array} \]

FPCore

k_m = (fabs.f64 k)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k_m)
 :precision binary64
 (* t_s (* (* l l) (/ -0.11666666666666667 t_m))))

C

k_m = fabs(k);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k_m) {
	return t_s * ((l * l) * (-0.11666666666666667 / t_m));
}

Fortran

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k_m
    code = t_s * ((l * l) * ((-0.11666666666666667d0) / t_m))
end function

Java

k_m = Math.abs(k);
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_m) {
	return t_s * ((l * l) * (-0.11666666666666667 / t_m));
}

Python

k_m = math.fabs(k)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k_m):
	return t_s * ((l * l) * (-0.11666666666666667 / t_m))

Julia

k_m = abs(k)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k_m)
	return Float64(t_s * Float64(Float64(l * l) * Float64(-0.11666666666666667 / t_m)))
end

MATLAB

k_m = abs(k);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, t_m, l, k_m)
	tmp = t_s * ((l * l) * (-0.11666666666666667 / t_m));
end

Wolfram

k_m = N[Abs[k], $MachinePrecision]
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$95$m_] := N[(t$95$s * N[(N[(l * l), $MachinePrecision] * N[(-0.11666666666666667 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

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

3. Simplified 38.8%

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

5. Taylor expanded in k around 0 36.3%

   \[\leadsto \color{blue}{\frac{{k}^{2} \cdot \left(-0.11666666666666667 \cdot \frac{{k}^{2}}{t} - 0.3333333333333333 \cdot \frac{1}{t}\right) + 2 \cdot \frac{1}{t}}{{k}^{4}}} \cdot \left(\ell \cdot \ell\right) \]

6. Taylor expanded in k around inf 19.8%

   \[\leadsto \color{blue}{\frac{-0.11666666666666667}{t}} \cdot \left(\ell \cdot \ell\right) \]

7. Final simplification 19.8%

   \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{-0.11666666666666667}{t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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))))