Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.6% → 88.5%

Time: 25.8s

Alternatives: 16

Speedup: 3.8×

• 40.8% 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: 35.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \sqrt{\sin k \cdot \tan k}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-207}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \left(k \cdot \frac{\sqrt{t\_m}}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;t\_m \leq 5.8 \cdot 10^{+186}:\\ \;\;\;\;2 \cdot \frac{1}{\sin k \cdot \left(\tan k \cdot {\left(\frac{k \cdot \frac{{t\_m}^{1.5}}{t\_m}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t\_2 \cdot \left(\sqrt{t\_m} \cdot \frac{k}{\ell}\right)\right)}^{2}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (sqrt (* (sin k) (tan k)))))
   (*
    t_s
    (if (<= t_m 1.15e-207)
      (/ 2.0 (pow (* t_2 (* k (/ (sqrt t_m) l))) 2.0))
      (if (<= t_m 5.8e+186)
        (*
         2.0
         (/
          1.0
          (* (sin k) (* (tan k) (pow (/ (* k (/ (pow t_m 1.5) t_m)) l) 2.0)))))
        (/ 2.0 (pow (* t_2 (* (sqrt t_m) (/ k l))) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = sqrt((sin(k) * tan(k)));
	double tmp;
	if (t_m <= 1.15e-207) {
		tmp = 2.0 / pow((t_2 * (k * (sqrt(t_m) / l))), 2.0);
	} else if (t_m <= 5.8e+186) {
		tmp = 2.0 * (1.0 / (sin(k) * (tan(k) * pow(((k * (pow(t_m, 1.5) / t_m)) / l), 2.0))));
	} else {
		tmp = 2.0 / pow((t_2 * (sqrt(t_m) * (k / l))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sqrt((sin(k) * tan(k)))
    if (t_m <= 1.15d-207) then
        tmp = 2.0d0 / ((t_2 * (k * (sqrt(t_m) / l))) ** 2.0d0)
    else if (t_m <= 5.8d+186) then
        tmp = 2.0d0 * (1.0d0 / (sin(k) * (tan(k) * (((k * ((t_m ** 1.5d0) / t_m)) / l) ** 2.0d0))))
    else
        tmp = 2.0d0 / ((t_2 * (sqrt(t_m) * (k / l))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.sqrt((Math.sin(k) * Math.tan(k)));
	double tmp;
	if (t_m <= 1.15e-207) {
		tmp = 2.0 / Math.pow((t_2 * (k * (Math.sqrt(t_m) / l))), 2.0);
	} else if (t_m <= 5.8e+186) {
		tmp = 2.0 * (1.0 / (Math.sin(k) * (Math.tan(k) * Math.pow(((k * (Math.pow(t_m, 1.5) / t_m)) / l), 2.0))));
	} else {
		tmp = 2.0 / Math.pow((t_2 * (Math.sqrt(t_m) * (k / l))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.sqrt((math.sin(k) * math.tan(k)))
	tmp = 0
	if t_m <= 1.15e-207:
		tmp = 2.0 / math.pow((t_2 * (k * (math.sqrt(t_m) / l))), 2.0)
	elif t_m <= 5.8e+186:
		tmp = 2.0 * (1.0 / (math.sin(k) * (math.tan(k) * math.pow(((k * (math.pow(t_m, 1.5) / t_m)) / l), 2.0))))
	else:
		tmp = 2.0 / math.pow((t_2 * (math.sqrt(t_m) * (k / l))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sqrt(Float64(sin(k) * tan(k)))
	tmp = 0.0
	if (t_m <= 1.15e-207)
		tmp = Float64(2.0 / (Float64(t_2 * Float64(k * Float64(sqrt(t_m) / l))) ^ 2.0));
	elseif (t_m <= 5.8e+186)
		tmp = Float64(2.0 * Float64(1.0 / Float64(sin(k) * Float64(tan(k) * (Float64(Float64(k * Float64((t_m ^ 1.5) / t_m)) / l) ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(t_2 * Float64(sqrt(t_m) * Float64(k / l))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sqrt((sin(k) * tan(k)));
	tmp = 0.0;
	if (t_m <= 1.15e-207)
		tmp = 2.0 / ((t_2 * (k * (sqrt(t_m) / l))) ^ 2.0);
	elseif (t_m <= 5.8e+186)
		tmp = 2.0 * (1.0 / (sin(k) * (tan(k) * (((k * ((t_m ^ 1.5) / t_m)) / l) ^ 2.0))));
	else
		tmp = 2.0 / ((t_2 * (sqrt(t_m) * (k / l))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.15e-207], N[(2.0 / N[Power[N[(t$95$2 * N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.8e+186], N[(2.0 * N[(1.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(t$95$2 * N[(N[Sqrt[t$95$m], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.15e-207

   1. Initial program 36.8%

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

      1. *-commutative 36.8%

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

      2. associate-/r* 36.8%

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

   3. Simplified 42.4%

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

      1. add-sqr-sqrt 11.3%

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

      2. pow2 11.3%

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

      3. sqrt-prod 3.2%

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

      4. sqrt-div 3.1%

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

      5. sqrt-pow1 3.1%

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

      6. metadata-eval 3.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 2.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 4.4%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 4.4%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 4.4%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 4.4%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 4.4%

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

      4. pow-prod-down 7.6%

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

      5. *-commutative 7.6%

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

   8. Applied egg-rr 7.6%

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

      1. *-lft-identity 7.6%

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

      2. associate-*l* 7.6%

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

   10. Simplified 7.6%

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

   11. Taylor expanded in t around 0 12.5%

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

       1. associate-*l/ 12.5%

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

       2. associate-/l* 12.5%

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

   13. Simplified 12.5%

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

   if 1.15e-207 < t < 5.8e186

   1. Initial program 43.3%

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

      1. *-commutative 43.3%

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

      2. associate-/r* 43.3%

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

   3. Simplified 46.2%

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

      1. add-sqr-sqrt 37.9%

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

      2. pow2 37.9%

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

      3. sqrt-prod 37.9%

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

      4. sqrt-div 39.3%

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

      5. sqrt-pow1 49.1%

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

      6. metadata-eval 49.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 26.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 55.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 55.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 55.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 55.7%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 55.7%

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

      4. pow-prod-down 68.9%

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

      5. *-commutative 68.9%

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

   8. Applied egg-rr 68.9%

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

      1. *-lft-identity 68.9%

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

      2. associate-*l* 68.9%

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

   10. Simplified 68.9%

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

       1. clear-num 68.9%

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

       2. inv-pow 68.9%

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

       3. unpow-prod-down 68.9%

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

       4. pow2 68.9%

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

       5. add-sqr-sqrt 85.5%

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

       6. *-commutative 85.5%

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

   12. Applied egg-rr 85.5%

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

       1. unpow-1 85.5%

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

       2. associate-/r/ 85.5%

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

       3. *-commutative 85.5%

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

       4. associate-*l* 85.5%

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

       5. associate-*l/ 89.6%

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

       6. associate-*r/ 94.9%

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

       7. *-commutative 94.9%

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

       8. associate-/l* 96.1%

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

   14. Simplified 96.1%

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

   if 5.8e186 < t

   1. Initial program 8.3%

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

      1. *-commutative 8.3%

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

      2. associate-/r* 8.3%

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

   3. Simplified 25.4%

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

      1. add-sqr-sqrt 20.8%

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

      2. pow2 20.8%

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

      3. sqrt-prod 20.8%

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

      4. sqrt-div 20.8%

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

      5. sqrt-pow1 25.0%

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

      6. metadata-eval 25.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 12.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 25.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 25.2%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 25.2%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 25.2%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 25.2%

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

      4. pow-prod-down 37.9%

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

      5. *-commutative 37.9%

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

   8. Applied egg-rr 37.9%

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

      1. *-lft-identity 37.9%

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

      2. associate-*l* 37.9%

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

   10. Simplified 37.9%

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

   11. Taylor expanded in t around 0 75.2%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-207}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{\sqrt{t}}{\ell}\right)\right)}^{2}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+186}:\\ \;\;\;\;2 \cdot \frac{1}{\sin k \cdot \left(\tan k \cdot {\left(\frac{k \cdot \frac{{t}^{1.5}}{t}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\sqrt{t} \cdot \frac{k}{\ell}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.15 \cdot 10^{-207} \lor \neg \left(t\_m \leq 1.45 \cdot 10^{+187}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{\sqrt{t\_m}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\sin k \cdot \left(\tan k \cdot {\left(\frac{k \cdot \frac{{t\_m}^{1.5}}{t\_m}}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 1.15e-207) (not (<= t_m 1.45e+187)))
    (/ 2.0 (pow (* (sqrt (* (sin k) (tan k))) (* k (/ (sqrt t_m) l))) 2.0))
    (*
     2.0
     (/
      1.0
      (* (sin k) (* (tan k) (pow (/ (* k (/ (pow t_m 1.5) t_m)) l) 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 1.15e-207) || !(t_m <= 1.45e+187)) {
		tmp = 2.0 / pow((sqrt((sin(k) * tan(k))) * (k * (sqrt(t_m) / l))), 2.0);
	} else {
		tmp = 2.0 * (1.0 / (sin(k) * (tan(k) * pow(((k * (pow(t_m, 1.5) / t_m)) / l), 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t_m <= 1.15d-207) .or. (.not. (t_m <= 1.45d+187))) then
        tmp = 2.0d0 / ((sqrt((sin(k) * tan(k))) * (k * (sqrt(t_m) / l))) ** 2.0d0)
    else
        tmp = 2.0d0 * (1.0d0 / (sin(k) * (tan(k) * (((k * ((t_m ** 1.5d0) / t_m)) / l) ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 1.15e-207) || !(t_m <= 1.45e+187)) {
		tmp = 2.0 / Math.pow((Math.sqrt((Math.sin(k) * Math.tan(k))) * (k * (Math.sqrt(t_m) / l))), 2.0);
	} else {
		tmp = 2.0 * (1.0 / (Math.sin(k) * (Math.tan(k) * Math.pow(((k * (Math.pow(t_m, 1.5) / t_m)) / l), 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (t_m <= 1.15e-207) or not (t_m <= 1.45e+187):
		tmp = 2.0 / math.pow((math.sqrt((math.sin(k) * math.tan(k))) * (k * (math.sqrt(t_m) / l))), 2.0)
	else:
		tmp = 2.0 * (1.0 / (math.sin(k) * (math.tan(k) * math.pow(((k * (math.pow(t_m, 1.5) / t_m)) / l), 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 1.15e-207) || !(t_m <= 1.45e+187))
		tmp = Float64(2.0 / (Float64(sqrt(Float64(sin(k) * tan(k))) * Float64(k * Float64(sqrt(t_m) / l))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64(sin(k) * Float64(tan(k) * (Float64(Float64(k * Float64((t_m ^ 1.5) / t_m)) / l) ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((t_m <= 1.15e-207) || ~((t_m <= 1.45e+187)))
		tmp = 2.0 / ((sqrt((sin(k) * tan(k))) * (k * (sqrt(t_m) / l))) ^ 2.0);
	else
		tmp = 2.0 * (1.0 / (sin(k) * (tan(k) * (((k * ((t_m ^ 1.5) / t_m)) / l) ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 1.15e-207], N[Not[LessEqual[t$95$m, 1.45e+187]], $MachinePrecision]], N[(2.0 / N[Power[N[(N[Sqrt[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(k * N[(N[Sqrt[t$95$m], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.15e-207 or 1.45e187 < t

   1. Initial program 33.1%

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

      1. *-commutative 33.1%

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

      2. associate-/r* 33.1%

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

   3. Simplified 40.2%

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

      1. add-sqr-sqrt 12.6%

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

      2. pow2 12.6%

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

      3. sqrt-prod 5.5%

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

      4. sqrt-div 5.5%

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

      5. sqrt-pow1 6.0%

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

      6. metadata-eval 6.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 3.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 7.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 7.1%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 7.1%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 7.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 7.1%

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

      4. pow-prod-down 11.5%

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

      5. *-commutative 11.5%

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

   8. Applied egg-rr 11.5%

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

      1. *-lft-identity 11.5%

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

      2. associate-*l* 11.5%

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

   10. Simplified 11.5%

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

   11. Taylor expanded in t around 0 20.7%

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

       1. associate-*l/ 20.2%

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

       2. associate-/l* 20.7%

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

   13. Simplified 20.7%

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

   if 1.15e-207 < t < 1.45e187

   1. Initial program 43.3%

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

      1. *-commutative 43.3%

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

      2. associate-/r* 43.3%

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

   3. Simplified 46.2%

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

      1. add-sqr-sqrt 37.9%

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

      2. pow2 37.9%

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

      3. sqrt-prod 37.9%

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

      4. sqrt-div 39.3%

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

      5. sqrt-pow1 49.1%

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

      6. metadata-eval 49.1%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 26.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 55.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 55.8%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 55.8%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 55.7%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 55.7%

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

      4. pow-prod-down 68.9%

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

      5. *-commutative 68.9%

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

   8. Applied egg-rr 68.9%

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

      1. *-lft-identity 68.9%

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

      2. associate-*l* 68.9%

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

   10. Simplified 68.9%

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

       1. clear-num 68.9%

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

       2. inv-pow 68.9%

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

       3. unpow-prod-down 68.9%

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

       4. pow2 68.9%

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

       5. add-sqr-sqrt 85.5%

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

       6. *-commutative 85.5%

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

   12. Applied egg-rr 85.5%

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

       1. unpow-1 85.5%

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

       2. associate-/r/ 85.5%

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

       3. *-commutative 85.5%

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

       4. associate-*l* 85.5%

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

       5. associate-*l/ 89.6%

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

       6. associate-*r/ 94.9%

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

       7. *-commutative 94.9%

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

       8. associate-/l* 96.1%

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

   14. Simplified 96.1%

       \[\leadsto \color{blue}{\frac{1}{\sin k \cdot \left(\tan k \cdot {\left(\frac{k \cdot \frac{{t}^{1.5}}{t}}{\ell}\right)}^{2}\right)} \cdot 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.15 \cdot 10^{-207} \lor \neg \left(t \leq 1.45 \cdot 10^{+187}\right):\\ \;\;\;\;\frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(k \cdot \frac{\sqrt{t}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\sin k \cdot \left(\tan k \cdot {\left(\frac{k \cdot \frac{{t}^{1.5}}{t}}{\ell}\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.7 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t\_m \leq 1.42 \cdot 10^{+189}:\\ \;\;\;\;2 \cdot \frac{1}{\sin k \cdot \left(\tan k \cdot {\left(\frac{k \cdot \frac{{t\_m}^{1.5}}{t\_m}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k}{t\_m}}{{k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.7e-223)
    (* (/ 2.0 (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (cos k))))) (* l l))
    (if (<= t_m 1.42e+189)
      (*
       2.0
       (/
        1.0
        (* (sin k) (* (tan k) (pow (/ (* k (/ (pow t_m 1.5) t_m)) l) 2.0)))))
      (* (* l l) (* (/ 2.0 (pow k 2.0)) (/ (/ (cos k) t_m) (pow k 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-223) {
		tmp = (2.0 / (pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / cos(k))))) * (l * l);
	} else if (t_m <= 1.42e+189) {
		tmp = 2.0 * (1.0 / (sin(k) * (tan(k) * pow(((k * (pow(t_m, 1.5) / t_m)) / l), 2.0))));
	} else {
		tmp = (l * l) * ((2.0 / pow(k, 2.0)) * ((cos(k) / t_m) / pow(k, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.7d-223) then
        tmp = (2.0d0 / ((k ** 2.0d0) * (t_m * ((sin(k) ** 2.0d0) / cos(k))))) * (l * l)
    else if (t_m <= 1.42d+189) then
        tmp = 2.0d0 * (1.0d0 / (sin(k) * (tan(k) * (((k * ((t_m ** 1.5d0) / t_m)) / l) ** 2.0d0))))
    else
        tmp = (l * l) * ((2.0d0 / (k ** 2.0d0)) * ((cos(k) / t_m) / (k ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.7e-223) {
		tmp = (2.0 / (Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))))) * (l * l);
	} else if (t_m <= 1.42e+189) {
		tmp = 2.0 * (1.0 / (Math.sin(k) * (Math.tan(k) * Math.pow(((k * (Math.pow(t_m, 1.5) / t_m)) / l), 2.0))));
	} else {
		tmp = (l * l) * ((2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / t_m) / Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.7e-223:
		tmp = (2.0 / (math.pow(k, 2.0) * (t_m * (math.pow(math.sin(k), 2.0) / math.cos(k))))) * (l * l)
	elif t_m <= 1.42e+189:
		tmp = 2.0 * (1.0 / (math.sin(k) * (math.tan(k) * math.pow(((k * (math.pow(t_m, 1.5) / t_m)) / l), 2.0))))
	else:
		tmp = (l * l) * ((2.0 / math.pow(k, 2.0)) * ((math.cos(k) / t_m) / math.pow(k, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.7e-223)
		tmp = Float64(Float64(2.0 / Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / cos(k))))) * Float64(l * l));
	elseif (t_m <= 1.42e+189)
		tmp = Float64(2.0 * Float64(1.0 / Float64(sin(k) * Float64(tan(k) * (Float64(Float64(k * Float64((t_m ^ 1.5) / t_m)) / l) ^ 2.0)))));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / t_m) / (k ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.7e-223)
		tmp = (2.0 / ((k ^ 2.0) * (t_m * ((sin(k) ^ 2.0) / cos(k))))) * (l * l);
	elseif (t_m <= 1.42e+189)
		tmp = 2.0 * (1.0 / (sin(k) * (tan(k) * (((k * ((t_m ^ 1.5) / t_m)) / l) ^ 2.0))));
	else
		tmp = (l * l) * ((2.0 / (k ^ 2.0)) * ((cos(k) / t_m) / (k ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.7e-223], N[(N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.42e+189], N[(2.0 * N[(1.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.6999999999999999e-223

   1. Initial program 37.2%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in t around 0 76.2%

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

      1. div-inv 76.2%

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

      2. associate-*r* 76.2%

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

   7. Applied egg-rr 76.2%

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

      1. associate-*r/ 76.2%

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

      2. metadata-eval 76.2%

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

      3. associate-/l* 76.2%

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

   9. Simplified 76.2%

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

       1. div-inv 76.2%

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

       2. *-commutative 76.2%

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

   11. Applied egg-rr 76.2%

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

       1. associate-*r/ 76.2%

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

       2. metadata-eval 76.2%

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

       3. *-commutative 76.2%

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

       4. associate-*l* 76.2%

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

   13. Simplified 76.2%

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

   if 1.6999999999999999e-223 < t < 1.42000000000000006e189

   1. Initial program 42.0%

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

      1. *-commutative 42.0%

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

      2. associate-/r* 42.0%

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

   3. Simplified 44.7%

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

      1. add-sqr-sqrt 37.1%

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

      2. pow2 37.1%

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

      3. sqrt-prod 37.1%

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

      4. sqrt-div 38.3%

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

      5. sqrt-pow1 47.3%

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

      6. metadata-eval 47.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 26.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 55.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 55.9%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 55.9%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 55.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 55.9%

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

      4. pow-prod-down 69.0%

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

      5. *-commutative 69.0%

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

   8. Applied egg-rr 69.0%

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

      1. *-lft-identity 69.0%

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

      2. associate-*l* 69.0%

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

   10. Simplified 69.0%

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

       1. clear-num 69.0%

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

       2. inv-pow 69.0%

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

       3. unpow-prod-down 69.0%

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

       4. pow2 69.0%

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

       5. add-sqr-sqrt 84.2%

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

       6. *-commutative 84.2%

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

   12. Applied egg-rr 84.2%

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

       1. unpow-1 84.2%

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

       2. associate-/r/ 84.2%

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

       3. *-commutative 84.2%

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

       4. associate-*l* 84.1%

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

       5. associate-*l/ 87.9%

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

       6. associate-*r/ 93.3%

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

       7. *-commutative 93.3%

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

       8. associate-/l* 95.5%

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

   14. Simplified 95.5%

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

   if 1.42000000000000006e189 < t

   1. Initial program 8.3%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in t around 0 75.5%

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

   6. Taylor expanded in k around inf 75.5%

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

      1. associate-*r* 75.5%

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

      2. associate-*r/ 75.5%

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

      3. associate-*r* 75.5%

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

      4. times-frac 75.5%

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

      5. associate-/r* 75.6%

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

   8. Simplified 75.6%

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

   9. Taylor expanded in k around 0 75.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+189}:\\ \;\;\;\;2 \cdot \frac{1}{\sin k \cdot \left(\tan k \cdot {\left(\frac{k \cdot \frac{{t}^{1.5}}{t}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k}{t}}{{k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\sin k}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.26 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t\_m \cdot \frac{t\_2}{\cos k}\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t\_m \leq 1.92 \cdot 10^{+161}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k}{t\_m}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k}{t\_m}}{t\_2}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (sin k) 2.0)))
   (*
    t_s
    (if (<= t_m 1.26e-223)
      (* (/ 2.0 (* (pow k 2.0) (* t_m (/ t_2 (cos k))))) (* l l))
      (if (<= t_m 1.92e+161)
        (/
         2.0
         (* (* (sin k) (tan k)) (pow (* (/ (pow t_m 1.5) l) (/ k t_m)) 2.0)))
        (* (* l l) (* (/ 2.0 (pow k 2.0)) (/ (/ (cos k) t_m) t_2))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow(sin(k), 2.0);
	double tmp;
	if (t_m <= 1.26e-223) {
		tmp = (2.0 / (pow(k, 2.0) * (t_m * (t_2 / cos(k))))) * (l * l);
	} else if (t_m <= 1.92e+161) {
		tmp = 2.0 / ((sin(k) * tan(k)) * pow(((pow(t_m, 1.5) / l) * (k / t_m)), 2.0));
	} else {
		tmp = (l * l) * ((2.0 / pow(k, 2.0)) * ((cos(k) / t_m) / t_2));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_2
    real(8) :: tmp
    t_2 = sin(k) ** 2.0d0
    if (t_m <= 1.26d-223) then
        tmp = (2.0d0 / ((k ** 2.0d0) * (t_m * (t_2 / cos(k))))) * (l * l)
    else if (t_m <= 1.92d+161) then
        tmp = 2.0d0 / ((sin(k) * tan(k)) * ((((t_m ** 1.5d0) / l) * (k / t_m)) ** 2.0d0))
    else
        tmp = (l * l) * ((2.0d0 / (k ** 2.0d0)) * ((cos(k) / t_m) / t_2))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (t_m <= 1.26e-223) {
		tmp = (2.0 / (Math.pow(k, 2.0) * (t_m * (t_2 / Math.cos(k))))) * (l * l);
	} else if (t_m <= 1.92e+161) {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * Math.pow(((Math.pow(t_m, 1.5) / l) * (k / t_m)), 2.0));
	} else {
		tmp = (l * l) * ((2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / t_m) / t_2));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if t_m <= 1.26e-223:
		tmp = (2.0 / (math.pow(k, 2.0) * (t_m * (t_2 / math.cos(k))))) * (l * l)
	elif t_m <= 1.92e+161:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * math.pow(((math.pow(t_m, 1.5) / l) * (k / t_m)), 2.0))
	else:
		tmp = (l * l) * ((2.0 / math.pow(k, 2.0)) * ((math.cos(k) / t_m) / t_2))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0
	tmp = 0.0
	if (t_m <= 1.26e-223)
		tmp = Float64(Float64(2.0 / Float64((k ^ 2.0) * Float64(t_m * Float64(t_2 / cos(k))))) * Float64(l * l));
	elseif (t_m <= 1.92e+161)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * (Float64(Float64((t_m ^ 1.5) / l) * Float64(k / t_m)) ^ 2.0)));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / t_m) / t_2)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (t_m <= 1.26e-223)
		tmp = (2.0 / ((k ^ 2.0) * (t_m * (t_2 / cos(k))))) * (l * l);
	elseif (t_m <= 1.92e+161)
		tmp = 2.0 / ((sin(k) * tan(k)) * ((((t_m ^ 1.5) / l) * (k / t_m)) ^ 2.0));
	else
		tmp = (l * l) * ((2.0 / (k ^ 2.0)) * ((cos(k) / t_m) / t_2));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.26e-223], N[(N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(t$95$2 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.92e+161], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 1.26000000000000009e-223

   1. Initial program 37.2%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in t around 0 76.2%

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

      1. div-inv 76.2%

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

      2. associate-*r* 76.2%

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

   7. Applied egg-rr 76.2%

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

      1. associate-*r/ 76.2%

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

      2. metadata-eval 76.2%

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

      3. associate-/l* 76.2%

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

   9. Simplified 76.2%

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

       1. div-inv 76.2%

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

       2. *-commutative 76.2%

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

   11. Applied egg-rr 76.2%

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

       1. associate-*r/ 76.2%

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

       2. metadata-eval 76.2%

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

       3. *-commutative 76.2%

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

       4. associate-*l* 76.2%

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

   13. Simplified 76.2%

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

   if 1.26000000000000009e-223 < t < 1.9200000000000001e161

   1. Initial program 45.5%

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

      1. *-commutative 45.5%

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

      2. associate-/r* 45.5%

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

   3. Simplified 48.2%

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

      1. add-sqr-sqrt 40.0%

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

      2. pow2 40.0%

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

      3. sqrt-prod 40.0%

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

      4. sqrt-div 41.3%

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

      5. sqrt-pow1 48.3%

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

      6. metadata-eval 48.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 27.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 57.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 57.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 57.5%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 57.5%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 57.5%

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

      4. pow-prod-down 69.3%

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

      5. *-commutative 69.3%

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

   8. Applied egg-rr 69.3%

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

      1. *-lft-identity 69.3%

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

      2. associate-*l* 69.3%

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

   10. Simplified 69.3%

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

       1. *-un-lft-identity 69.3%

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

       2. unpow-prod-down 69.2%

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

       3. pow2 69.2%

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

       4. add-sqr-sqrt 85.6%

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

       5. *-commutative 85.6%

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

   12. Applied egg-rr 85.6%

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

       1. *-lft-identity 85.6%

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

       2. *-commutative 85.6%

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

   14. Simplified 85.6%

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

   if 1.9200000000000001e161 < t

   1. Initial program 6.7%

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

   3. Simplified 23.9%

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

   5. Taylor expanded in t around 0 77.0%

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

   6. Taylor expanded in k around inf 77.0%

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

      1. associate-*r* 77.0%

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

      2. associate-*r/ 77.0%

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

      3. associate-*r* 77.0%

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

      4. times-frac 77.1%

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

      5. associate-/r* 77.1%

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

   8. Simplified 77.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.26 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t \leq 1.92 \cdot 10^{+161}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.6 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t\_m \leq 1.42 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k \cdot \frac{{t\_m}^{1.5}}{t\_m}}{\ell}\right)}^{2}}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k}{t\_m}}{{k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.6e-223)
    (* (/ 2.0 (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (cos k))))) (* l l))
    (if (<= t_m 1.42e+189)
      (/
       (/ 2.0 (pow (/ (* k (/ (pow t_m 1.5) t_m)) l) 2.0))
       (* (sin k) (tan k)))
      (* (* l l) (* (/ 2.0 (pow k 2.0)) (/ (/ (cos k) t_m) (pow k 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.6e-223) {
		tmp = (2.0 / (pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / cos(k))))) * (l * l);
	} else if (t_m <= 1.42e+189) {
		tmp = (2.0 / pow(((k * (pow(t_m, 1.5) / t_m)) / l), 2.0)) / (sin(k) * tan(k));
	} else {
		tmp = (l * l) * ((2.0 / pow(k, 2.0)) * ((cos(k) / t_m) / pow(k, 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2.6d-223) then
        tmp = (2.0d0 / ((k ** 2.0d0) * (t_m * ((sin(k) ** 2.0d0) / cos(k))))) * (l * l)
    else if (t_m <= 1.42d+189) then
        tmp = (2.0d0 / (((k * ((t_m ** 1.5d0) / t_m)) / l) ** 2.0d0)) / (sin(k) * tan(k))
    else
        tmp = (l * l) * ((2.0d0 / (k ** 2.0d0)) * ((cos(k) / t_m) / (k ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2.6e-223) {
		tmp = (2.0 / (Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))))) * (l * l);
	} else if (t_m <= 1.42e+189) {
		tmp = (2.0 / Math.pow(((k * (Math.pow(t_m, 1.5) / t_m)) / l), 2.0)) / (Math.sin(k) * Math.tan(k));
	} else {
		tmp = (l * l) * ((2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / t_m) / Math.pow(k, 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2.6e-223:
		tmp = (2.0 / (math.pow(k, 2.0) * (t_m * (math.pow(math.sin(k), 2.0) / math.cos(k))))) * (l * l)
	elif t_m <= 1.42e+189:
		tmp = (2.0 / math.pow(((k * (math.pow(t_m, 1.5) / t_m)) / l), 2.0)) / (math.sin(k) * math.tan(k))
	else:
		tmp = (l * l) * ((2.0 / math.pow(k, 2.0)) * ((math.cos(k) / t_m) / math.pow(k, 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2.6e-223)
		tmp = Float64(Float64(2.0 / Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / cos(k))))) * Float64(l * l));
	elseif (t_m <= 1.42e+189)
		tmp = Float64(Float64(2.0 / (Float64(Float64(k * Float64((t_m ^ 1.5) / t_m)) / l) ^ 2.0)) / Float64(sin(k) * tan(k)));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / t_m) / (k ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2.6e-223)
		tmp = (2.0 / ((k ^ 2.0) * (t_m * ((sin(k) ^ 2.0) / cos(k))))) * (l * l);
	elseif (t_m <= 1.42e+189)
		tmp = (2.0 / (((k * ((t_m ^ 1.5) / t_m)) / l) ^ 2.0)) / (sin(k) * tan(k));
	else
		tmp = (l * l) * ((2.0 / (k ^ 2.0)) * ((cos(k) / t_m) / (k ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2.6e-223], N[(N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.42e+189], N[(N[(2.0 / N[Power[N[(N[(k * N[(N[Power[t$95$m, 1.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.6e-223

   1. Initial program 37.2%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in t around 0 76.2%

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

      1. div-inv 76.2%

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

      2. associate-*r* 76.2%

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

   7. Applied egg-rr 76.2%

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

      1. associate-*r/ 76.2%

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

      2. metadata-eval 76.2%

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

      3. associate-/l* 76.2%

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

   9. Simplified 76.2%

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

       1. div-inv 76.2%

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

       2. *-commutative 76.2%

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

   11. Applied egg-rr 76.2%

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

       1. associate-*r/ 76.2%

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

       2. metadata-eval 76.2%

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

       3. *-commutative 76.2%

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

       4. associate-*l* 76.2%

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

   13. Simplified 76.2%

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

   if 2.6e-223 < t < 1.42000000000000006e189

   1. Initial program 42.0%

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

      1. *-commutative 42.0%

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

      2. associate-/r* 42.0%

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

   3. Simplified 44.7%

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

      1. add-sqr-sqrt 37.1%

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

      2. pow2 37.1%

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

      3. sqrt-prod 37.1%

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

      4. sqrt-div 38.3%

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

      5. sqrt-pow1 47.3%

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

      6. metadata-eval 47.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 26.8%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 55.9%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 55.9%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 55.9%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 55.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 55.9%

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

      4. pow-prod-down 69.0%

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

      5. *-commutative 69.0%

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

   8. Applied egg-rr 69.0%

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

      1. *-lft-identity 69.0%

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

      2. associate-*l* 69.0%

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

   10. Simplified 69.0%

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

       1. *-un-lft-identity 69.0%

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

       2. unpow-prod-down 69.0%

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

       3. pow2 69.0%

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

       4. add-sqr-sqrt 84.2%

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

       5. *-commutative 84.2%

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

   12. Applied egg-rr 84.2%

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

       1. *-lft-identity 84.2%

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

       2. *-commutative 84.2%

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

       3. associate-/r* 84.1%

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

       4. associate-*l/ 87.9%

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

       5. associate-*r/ 92.1%

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

       6. *-commutative 92.1%

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

       7. associate-/l* 94.4%

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

       8. *-commutative 94.4%

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

   14. Simplified 94.4%

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

   if 1.42000000000000006e189 < t

   1. Initial program 8.3%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in t around 0 75.5%

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

   6. Taylor expanded in k around inf 75.5%

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

      1. associate-*r* 75.5%

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

      2. associate-*r/ 75.5%

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

      3. associate-*r* 75.5%

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

      4. times-frac 75.5%

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

      5. associate-/r* 75.6%

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

   8. Simplified 75.6%

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

   9. Taylor expanded in k around 0 75.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-223}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \cdot \left(\ell \cdot \ell\right)\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{2}{{\left(\frac{k \cdot \frac{{t}^{1.5}}{t}}{\ell}\right)}^{2}}}{\sin k \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k}{t}}{{k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k}{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\cos k \cdot \frac{2}{{\sin k}^{2} \cdot \left(t\_m \cdot {k}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k (* (/ (pow t_m 1.5) l) (/ k t_m))) 2.0))
    (*
     (* l l)
     (* (cos k) (/ 2.0 (* (pow (sin k) 2.0) (* t_m (pow k 2.0)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k * ((pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * (cos(k) * (2.0 / (pow(sin(k), 2.0) * (t_m * pow(k, 2.0)))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k * (((t_m ** 1.5d0) / l) * (k / t_m))) ** 2.0d0)
    else
        tmp = (l * l) * (cos(k) * (2.0d0 / ((sin(k) ** 2.0d0) * (t_m * (k ** 2.0d0)))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k * ((Math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * (Math.cos(k) * (2.0 / (Math.pow(Math.sin(k), 2.0) * (t_m * Math.pow(k, 2.0)))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k * ((math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0)
	else:
		tmp = (l * l) * (math.cos(k) * (2.0 / (math.pow(math.sin(k), 2.0) * (t_m * math.pow(k, 2.0)))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64((t_m ^ 1.5) / l) * Float64(k / t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(cos(k) * Float64(2.0 / Float64((sin(k) ^ 2.0) * Float64(t_m * (k ^ 2.0))))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k * (((t_m ^ 1.5) / l) * (k / t_m))) ^ 2.0);
	else
		tmp = (l * l) * (cos(k) * (2.0 / ((sin(k) ^ 2.0) * (t_m * (k ^ 2.0)))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(2.0 / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.7%

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

      1. *-commutative 19.7%

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

      2. associate-/r* 19.7%

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

   3. Simplified 26.5%

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

      1. add-sqr-sqrt 9.9%

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

      2. pow2 9.9%

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

      3. sqrt-prod 6.6%

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

      4. sqrt-div 6.6%

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

      5. sqrt-pow1 6.7%

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

      6. metadata-eval 6.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 8.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 18.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 18.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 18.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 17.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 17.9%

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

      4. pow-prod-down 23.1%

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

      5. *-commutative 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. *-lft-identity 23.1%

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

      2. associate-*l* 23.1%

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

   10. Simplified 23.1%

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

   11. Taylor expanded in k around 0 36.2%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.1%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in t around 0 80.2%

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

      1. associate-/r/ 80.2%

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

      2. associate-*r* 80.2%

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

   7. Applied egg-rr 80.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\cos k \cdot \frac{2}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k}{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k}{t\_m}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k (* (/ (pow t_m 1.5) l) (/ k t_m))) 2.0))
    (*
     (* l l)
     (* (/ 2.0 (pow k 2.0)) (/ (/ (cos k) t_m) (pow (sin k) 2.0)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k * ((pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / pow(k, 2.0)) * ((cos(k) / t_m) / pow(sin(k), 2.0)));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k * (((t_m ** 1.5d0) / l) * (k / t_m))) ** 2.0d0)
    else
        tmp = (l * l) * ((2.0d0 / (k ** 2.0d0)) * ((cos(k) / t_m) / (sin(k) ** 2.0d0)))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k * ((Math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 / Math.pow(k, 2.0)) * ((Math.cos(k) / t_m) / Math.pow(Math.sin(k), 2.0)));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k * ((math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0)
	else:
		tmp = (l * l) * ((2.0 / math.pow(k, 2.0)) * ((math.cos(k) / t_m) / math.pow(math.sin(k), 2.0)))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64((t_m ^ 1.5) / l) * Float64(k / t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / (k ^ 2.0)) * Float64(Float64(cos(k) / t_m) / (sin(k) ^ 2.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k * (((t_m ^ 1.5) / l) * (k / t_m))) ^ 2.0);
	else
		tmp = (l * l) * ((2.0 / (k ^ 2.0)) * ((cos(k) / t_m) / (sin(k) ^ 2.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.7%

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

      1. *-commutative 19.7%

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

      2. associate-/r* 19.7%

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

   3. Simplified 26.5%

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

      1. add-sqr-sqrt 9.9%

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

      2. pow2 9.9%

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

      3. sqrt-prod 6.6%

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

      4. sqrt-div 6.6%

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

      5. sqrt-pow1 6.7%

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

      6. metadata-eval 6.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 8.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 18.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 18.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 18.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 17.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 17.9%

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

      4. pow-prod-down 23.1%

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

      5. *-commutative 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. *-lft-identity 23.1%

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

      2. associate-*l* 23.1%

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

   10. Simplified 23.1%

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

   11. Taylor expanded in k around 0 36.2%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.1%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in t around 0 80.2%

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

   6. Taylor expanded in k around inf 80.2%

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

      1. associate-*r* 80.2%

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

      2. associate-*r/ 80.2%

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

      3. associate-*r* 80.2%

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

      4. times-frac 80.2%

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

      5. associate-/r* 80.2%

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

   8. Simplified 80.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k}{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(t\_m \cdot {k}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k (* (/ (pow t_m 1.5) l) (/ k t_m))) 2.0))
    (*
     (* l l)
     (/ 2.0 (* (/ (pow (sin k) 2.0) (cos k)) (* t_m (pow k 2.0))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k * ((pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((pow(sin(k), 2.0) / cos(k)) * (t_m * pow(k, 2.0))));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k * (((t_m ** 1.5d0) / l) * (k / t_m))) ** 2.0d0)
    else
        tmp = (l * l) * (2.0d0 / (((sin(k) ** 2.0d0) / cos(k)) * (t_m * (k ** 2.0d0))))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k * ((Math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * (2.0 / ((Math.pow(Math.sin(k), 2.0) / Math.cos(k)) * (t_m * Math.pow(k, 2.0))));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k * ((math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0)
	else:
		tmp = (l * l) * (2.0 / ((math.pow(math.sin(k), 2.0) / math.cos(k)) * (t_m * math.pow(k, 2.0))))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64((t_m ^ 1.5) / l) * Float64(k / t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(Float64((sin(k) ^ 2.0) / cos(k)) * Float64(t_m * (k ^ 2.0)))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k * (((t_m ^ 1.5) / l) * (k / t_m))) ^ 2.0);
	else
		tmp = (l * l) * (2.0 / (((sin(k) ^ 2.0) / cos(k)) * (t_m * (k ^ 2.0))));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.7%

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

      1. *-commutative 19.7%

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

      2. associate-/r* 19.7%

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

   3. Simplified 26.5%

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

      1. add-sqr-sqrt 9.9%

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

      2. pow2 9.9%

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

      3. sqrt-prod 6.6%

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

      4. sqrt-div 6.6%

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

      5. sqrt-pow1 6.7%

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

      6. metadata-eval 6.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 8.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 18.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 18.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 18.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 17.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 17.9%

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

      4. pow-prod-down 23.1%

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

      5. *-commutative 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. *-lft-identity 23.1%

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

      2. associate-*l* 23.1%

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

   10. Simplified 23.1%

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

   11. Taylor expanded in k around 0 36.2%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.1%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in t around 0 80.2%

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

      1. div-inv 80.2%

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

      2. associate-*r* 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. associate-*r/ 80.2%

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

      2. metadata-eval 80.2%

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

      3. associate-/l* 80.2%

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

   9. Simplified 80.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\frac{{\sin k}^{2}}{\cos k} \cdot \left(t \cdot {k}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k}{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t\_m \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k (* (/ (pow t_m 1.5) l) (/ k t_m))) 2.0))
    (*
     (/ 2.0 (* (pow k 2.0) (* t_m (/ (pow (sin k) 2.0) (cos k)))))
     (* l l)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k * ((pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (2.0 / (pow(k, 2.0) * (t_m * (pow(sin(k), 2.0) / cos(k))))) * (l * l);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k * (((t_m ** 1.5d0) / l) * (k / t_m))) ** 2.0d0)
    else
        tmp = (2.0d0 / ((k ** 2.0d0) * (t_m * ((sin(k) ** 2.0d0) / cos(k))))) * (l * l)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k * ((Math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (2.0 / (Math.pow(k, 2.0) * (t_m * (Math.pow(Math.sin(k), 2.0) / Math.cos(k))))) * (l * l);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k * ((math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0)
	else:
		tmp = (2.0 / (math.pow(k, 2.0) * (t_m * (math.pow(math.sin(k), 2.0) / math.cos(k))))) * (l * l)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64((t_m ^ 1.5) / l) * Float64(k / t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(2.0 / Float64((k ^ 2.0) * Float64(t_m * Float64((sin(k) ^ 2.0) / cos(k))))) * Float64(l * l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k * (((t_m ^ 1.5) / l) * (k / t_m))) ^ 2.0);
	else
		tmp = (2.0 / ((k ^ 2.0) * (t_m * ((sin(k) ^ 2.0) / cos(k))))) * (l * l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.7%

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

      1. *-commutative 19.7%

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

      2. associate-/r* 19.7%

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

   3. Simplified 26.5%

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

      1. add-sqr-sqrt 9.9%

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

      2. pow2 9.9%

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

      3. sqrt-prod 6.6%

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

      4. sqrt-div 6.6%

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

      5. sqrt-pow1 6.7%

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

      6. metadata-eval 6.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 8.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 18.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 18.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 18.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 17.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 17.9%

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

      4. pow-prod-down 23.1%

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

      5. *-commutative 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. *-lft-identity 23.1%

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

      2. associate-*l* 23.1%

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

   10. Simplified 23.1%

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

   11. Taylor expanded in k around 0 36.2%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.1%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in t around 0 80.2%

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

      1. div-inv 80.2%

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

      2. associate-*r* 80.2%

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

   7. Applied egg-rr 80.2%

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

      1. associate-*r/ 80.2%

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

      2. metadata-eval 80.2%

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

      3. associate-/l* 80.2%

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

   9. Simplified 80.2%

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

       1. div-inv 80.2%

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

       2. *-commutative 80.2%

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

   11. Applied egg-rr 80.2%

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

       1. associate-*r/ 80.2%

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

       2. metadata-eval 80.2%

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

       3. *-commutative 80.2%

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

       4. associate-*l* 80.2%

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

   13. Simplified 80.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{\sin k}^{2}}{\cos k}\right)} \cdot \left(\ell \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k}{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \frac{\cos k}{t\_m}}{{\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= (* l l) 0.0)
    (/ 2.0 (pow (* k (* (/ (pow t_m 1.5) l) (/ k t_m))) 2.0))
    (* (* l l) (/ (* 2.0 (/ (cos k) t_m)) (pow (* k (sin k)) 2.0))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / pow((k * ((pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 * (cos(k) / t_m)) / pow((k * sin(k)), 2.0));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / ((k * (((t_m ** 1.5d0) / l) * (k / t_m))) ** 2.0d0)
    else
        tmp = (l * l) * ((2.0d0 * (cos(k) / t_m)) / ((k * sin(k)) ** 2.0d0))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / Math.pow((k * ((Math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 * (Math.cos(k) / t_m)) / Math.pow((k * Math.sin(k)), 2.0));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / math.pow((k * ((math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0)
	else:
		tmp = (l * l) * ((2.0 * (math.cos(k) / t_m)) / math.pow((k * math.sin(k)), 2.0))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64((t_m ^ 1.5) / l) * Float64(k / t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * Float64(cos(k) / t_m)) / (Float64(k * sin(k)) ^ 2.0)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / ((k * (((t_m ^ 1.5) / l) * (k / t_m))) ^ 2.0);
	else
		tmp = (l * l) * ((2.0 * (cos(k) / t_m)) / ((k * sin(k)) ^ 2.0));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[Power[N[(k * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 19.7%

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

      1. *-commutative 19.7%

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

      2. associate-/r* 19.7%

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

   3. Simplified 26.5%

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

      1. add-sqr-sqrt 9.9%

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

      2. pow2 9.9%

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

      3. sqrt-prod 6.6%

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

      4. sqrt-div 6.6%

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

      5. sqrt-pow1 6.7%

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

      6. metadata-eval 6.7%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 8.2%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 18.0%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 18.0%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 18.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 17.9%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 17.9%

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

      4. pow-prod-down 23.1%

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

      5. *-commutative 23.1%

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

   8. Applied egg-rr 23.1%

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

      1. *-lft-identity 23.1%

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

      2. associate-*l* 23.1%

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

   10. Simplified 23.1%

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

   11. Taylor expanded in k around 0 36.2%

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

   if 0.0 < (*.f64 l l)

   1. Initial program 41.1%

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

   3. Simplified 48.7%

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

   5. Taylor expanded in t around 0 80.2%

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

   6. Taylor expanded in k around inf 80.2%

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

      1. associate-*r* 80.2%

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

      2. associate-*r/ 80.2%

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

      3. associate-*r* 80.2%

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

      4. times-frac 80.2%

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

      5. associate-/r* 80.2%

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

   8. Simplified 80.2%

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

      1. frac-times 79.1%

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

      2. pow-prod-down 79.1%

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

   10. Applied egg-rr 79.1%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot \frac{\cos k}{t}}{{\left(k \cdot \sin k\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.1 \cdot 10^{-223} \lor \neg \left(t\_m \leq 6.6 \cdot 10^{+160}\right):\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{1}{t\_m \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k}{t\_m}\right)\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (or (<= t_m 3.1e-223) (not (<= t_m 6.6e+160)))
    (* (* l l) (* (/ 2.0 (pow k 2.0)) (/ 1.0 (* t_m (pow k 2.0)))))
    (/ 2.0 (pow (* k (* (/ (pow t_m 1.5) l) (/ k t_m))) 2.0)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 3.1e-223) || !(t_m <= 6.6e+160)) {
		tmp = (l * l) * ((2.0 / pow(k, 2.0)) * (1.0 / (t_m * pow(k, 2.0))));
	} else {
		tmp = 2.0 / pow((k * ((pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t_m <= 3.1d-223) .or. (.not. (t_m <= 6.6d+160))) then
        tmp = (l * l) * ((2.0d0 / (k ** 2.0d0)) * (1.0d0 / (t_m * (k ** 2.0d0))))
    else
        tmp = 2.0d0 / ((k * (((t_m ** 1.5d0) / l) * (k / t_m))) ** 2.0d0)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if ((t_m <= 3.1e-223) || !(t_m <= 6.6e+160)) {
		tmp = (l * l) * ((2.0 / Math.pow(k, 2.0)) * (1.0 / (t_m * Math.pow(k, 2.0))));
	} else {
		tmp = 2.0 / Math.pow((k * ((Math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if (t_m <= 3.1e-223) or not (t_m <= 6.6e+160):
		tmp = (l * l) * ((2.0 / math.pow(k, 2.0)) * (1.0 / (t_m * math.pow(k, 2.0))))
	else:
		tmp = 2.0 / math.pow((k * ((math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if ((t_m <= 3.1e-223) || !(t_m <= 6.6e+160))
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 / (k ^ 2.0)) * Float64(1.0 / Float64(t_m * (k ^ 2.0)))));
	else
		tmp = Float64(2.0 / (Float64(k * Float64(Float64((t_m ^ 1.5) / l) * Float64(k / t_m))) ^ 2.0));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if ((t_m <= 3.1e-223) || ~((t_m <= 6.6e+160)))
		tmp = (l * l) * ((2.0 / (k ^ 2.0)) * (1.0 / (t_m * (k ^ 2.0))));
	else
		tmp = 2.0 / ((k * (((t_m ^ 1.5) / l) * (k / t_m))) ^ 2.0);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[Or[LessEqual[t$95$m, 3.1e-223], N[Not[LessEqual[t$95$m, 6.6e+160]], $MachinePrecision]], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$m * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[N[(k * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.10000000000000018e-223 or 6.5999999999999994e160 < t

   1. Initial program 32.1%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in t around 0 76.3%

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

   6. Taylor expanded in k around inf 76.3%

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

      1. associate-*r* 76.3%

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

      2. associate-*r/ 76.3%

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

      3. associate-*r* 76.3%

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

      4. times-frac 76.3%

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

      5. associate-/r* 76.3%

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

   8. Simplified 76.3%

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

   9. Taylor expanded in k around 0 68.4%

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

   if 3.10000000000000018e-223 < t < 6.5999999999999994e160

   1. Initial program 45.5%

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

      1. *-commutative 45.5%

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

      2. associate-/r* 45.5%

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

   3. Simplified 48.2%

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

      1. add-sqr-sqrt 40.0%

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

      2. pow2 40.0%

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

      3. sqrt-prod 40.0%

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

      4. sqrt-div 41.3%

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

      5. sqrt-pow1 48.3%

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

      6. metadata-eval 48.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 27.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 57.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 57.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 57.5%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 57.5%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 57.5%

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

      4. pow-prod-down 69.3%

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

      5. *-commutative 69.3%

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

   8. Applied egg-rr 69.3%

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

      1. *-lft-identity 69.3%

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

      2. associate-*l* 69.3%

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

   10. Simplified 69.3%

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

   11. Taylor expanded in k around 0 74.0%

       \[\leadsto \frac{2}{{\left(\color{blue}{k} \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.1 \cdot 10^{-223} \lor \neg \left(t \leq 6.6 \cdot 10^{+160}\right):\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\frac{2}{{k}^{2}} \cdot \frac{1}{t \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3 \cdot 10^{-223}:\\ \;\;\;\;{k}^{-4} \cdot \left(\frac{2}{t\_m} \cdot {\ell}^{2}\right)\\ \mathbf{elif}\;t\_m \leq 2.1 \cdot 10^{+161}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t\_m}^{1.5}}{\ell} \cdot \frac{k}{t\_m}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-4}}{t\_m}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 3e-223)
    (* (pow k -4.0) (* (/ 2.0 t_m) (pow l 2.0)))
    (if (<= t_m 2.1e+161)
      (/ 2.0 (pow (* k (* (/ (pow t_m 1.5) l) (/ k t_m))) 2.0))
      (* (* l l) (/ (* 2.0 (pow k -4.0)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-223) {
		tmp = pow(k, -4.0) * ((2.0 / t_m) * pow(l, 2.0));
	} else if (t_m <= 2.1e+161) {
		tmp = 2.0 / pow((k * ((pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 * pow(k, -4.0)) / t_m);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 3d-223) then
        tmp = (k ** (-4.0d0)) * ((2.0d0 / t_m) * (l ** 2.0d0))
    else if (t_m <= 2.1d+161) then
        tmp = 2.0d0 / ((k * (((t_m ** 1.5d0) / l) * (k / t_m))) ** 2.0d0)
    else
        tmp = (l * l) * ((2.0d0 * (k ** (-4.0d0))) / t_m)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 3e-223) {
		tmp = Math.pow(k, -4.0) * ((2.0 / t_m) * Math.pow(l, 2.0));
	} else if (t_m <= 2.1e+161) {
		tmp = 2.0 / Math.pow((k * ((Math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0);
	} else {
		tmp = (l * l) * ((2.0 * Math.pow(k, -4.0)) / t_m);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 3e-223:
		tmp = math.pow(k, -4.0) * ((2.0 / t_m) * math.pow(l, 2.0))
	elif t_m <= 2.1e+161:
		tmp = 2.0 / math.pow((k * ((math.pow(t_m, 1.5) / l) * (k / t_m))), 2.0)
	else:
		tmp = (l * l) * ((2.0 * math.pow(k, -4.0)) / t_m)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 3e-223)
		tmp = Float64((k ^ -4.0) * Float64(Float64(2.0 / t_m) * (l ^ 2.0)));
	elseif (t_m <= 2.1e+161)
		tmp = Float64(2.0 / (Float64(k * Float64(Float64((t_m ^ 1.5) / l) * Float64(k / t_m))) ^ 2.0));
	else
		tmp = Float64(Float64(l * l) * Float64(Float64(2.0 * (k ^ -4.0)) / t_m));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 3e-223)
		tmp = (k ^ -4.0) * ((2.0 / t_m) * (l ^ 2.0));
	elseif (t_m <= 2.1e+161)
		tmp = 2.0 / ((k * (((t_m ^ 1.5) / l) * (k / t_m))) ^ 2.0);
	else
		tmp = (l * l) * ((2.0 * (k ^ -4.0)) / t_m);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 3e-223], N[(N[Power[k, -4.0], $MachinePrecision] * N[(N[(2.0 / t$95$m), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.1e+161], N[(2.0 / N[Power[N[(k * N[(N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision] * N[(k / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 * N[Power[k, -4.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 2.99999999999999991e-223

   1. Initial program 37.2%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in k around 0 63.9%

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

      1. *-un-lft-identity 63.9%

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

      2. associate-/r* 63.5%

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

   7. Applied egg-rr 63.5%

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

      1. div-inv 63.5%

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

      2. div-inv 63.5%

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

      3. pow-flip 64.1%

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

      4. metadata-eval 64.1%

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

   9. Applied egg-rr 64.1%

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

       1. *-commutative 64.1%

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

       2. associate-*r* 64.1%

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

       3. associate-*l/ 64.1%

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

       4. metadata-eval 64.1%

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

   11. Simplified 64.1%

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

       1. pow1 64.1%

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

       2. *-un-lft-identity 64.1%

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

       3. pow2 64.1%

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

   13. Applied egg-rr 64.1%

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

       1. unpow1 64.1%

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

       2. *-commutative 64.1%

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

       3. associate-*l* 64.6%

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

   15. Simplified 64.6%

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

   if 2.99999999999999991e-223 < t < 2.1e161

   1. Initial program 45.5%

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

      1. *-commutative 45.5%

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

      2. associate-/r* 45.5%

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

   3. Simplified 48.2%

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

      1. add-sqr-sqrt 40.0%

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

      2. pow2 40.0%

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

      3. sqrt-prod 40.0%

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

      4. sqrt-div 41.3%

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

      5. sqrt-pow1 48.3%

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

      6. metadata-eval 48.3%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      7. sqrt-prod 27.6%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

      8. add-sqr-sqrt 57.5%

         \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}} \]

   6. Applied egg-rr 57.5%

      \[\leadsto \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 57.5%

         \[\leadsto \color{blue}{1 \cdot \frac{\frac{2}{{\left(\frac{k}{t}\right)}^{2} + 0}}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2}}} \]

      2. associate-/l/ 57.5%

         \[\leadsto 1 \cdot \color{blue}{\frac{2}{{\left(\frac{{t}^{1.5}}{\ell} \cdot \sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)}} \]

      3. +-rgt-identity 57.5%

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

      4. pow-prod-down 69.3%

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

      5. *-commutative 69.3%

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

   8. Applied egg-rr 69.3%

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

      1. *-lft-identity 69.3%

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

      2. associate-*l* 69.3%

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

   10. Simplified 69.3%

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

   11. Taylor expanded in k around 0 74.0%

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

   if 2.1e161 < t

   1. Initial program 6.7%

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

   3. Simplified 23.9%

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

   5. Taylor expanded in k around 0 74.1%

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

      1. *-un-lft-identity 74.1%

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

      2. associate-/r* 74.0%

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

   7. Applied egg-rr 74.0%

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

      1. div-inv 74.0%

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

      2. div-inv 74.0%

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

      3. pow-flip 74.0%

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

      4. metadata-eval 74.0%

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

   9. Applied egg-rr 74.0%

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

       1. associate-*r/ 74.1%

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

       2. *-rgt-identity 74.1%

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

   11. Simplified 74.1%

       \[\leadsto \left(1 \cdot \color{blue}{\frac{2 \cdot {k}^{-4}}{t}}\right) \cdot \left(\ell \cdot \ell\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-223}:\\ \;\;\;\;{k}^{-4} \cdot \left(\frac{2}{t} \cdot {\ell}^{2}\right)\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+161}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{k}{t}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2 \cdot {k}^{-4}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

3. Simplified 43.8%

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

5. Taylor expanded in k around 0 64.8%

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

6. Taylor expanded in k around 0 64.9%

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

   1. associate-/r* 66.0%

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

8. Simplified 66.0%

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

9. Final simplification 66.0%

   \[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{4}}}{t} \]
10. Add Preprocessing

Alternative 14: 62.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

3. Simplified 43.8%

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

5. Taylor expanded in k around 0 64.8%

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

6. Final simplification 64.8%

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

Alternative 15: 62.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

3. Simplified 43.8%

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

5. Taylor expanded in k around 0 64.8%

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

   1. *-un-lft-identity 64.8%

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

   2. associate-/r* 64.6%

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

7. Applied egg-rr 64.6%

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

   1. div-inv 64.6%

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

   2. div-inv 64.6%

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

   3. pow-flip 65.0%

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

   4. metadata-eval 65.0%

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

9. Applied egg-rr 65.0%

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

    1. *-commutative 65.0%

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

    2. associate-*r* 65.0%

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

    3. associate-*l/ 65.0%

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

    4. metadata-eval 65.0%

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

11. Simplified 65.0%

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

12. Final simplification 65.0%

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

Alternative 16: 62.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

3. Simplified 43.8%

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

5. Taylor expanded in k around 0 64.8%

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

   1. *-un-lft-identity 64.8%

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

   2. associate-/r* 64.6%

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

7. Applied egg-rr 64.6%

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

   1. div-inv 64.6%

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

   2. div-inv 64.6%

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

   3. pow-flip 65.0%

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

   4. metadata-eval 65.0%

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

9. Applied egg-rr 65.0%

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

    1. associate-*r/ 65.0%

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

    2. *-rgt-identity 65.0%

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

11. Simplified 65.0%

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

12. Final simplification 65.0%

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

Reproduce

herbie shell --seed 2024131 
(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))))