Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.0% → 90.7%

Time: 22.8s

Alternatives: 6

Speedup: 2.0×

• 35.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (pow (/ k l) 2.0) (/ (pow (sin k) 2.0) (/ (cos k) t)))))

C

double code(double t, double l, double k) {
	return 2.0 / (pow((k / l), 2.0) * (pow(sin(k), 2.0) / (cos(k) / t)));
}

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 / (((k / l) ** 2.0d0) * ((sin(k) ** 2.0d0) / (cos(k) / t)))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 / (math.pow((k / l), 2.0) * (math.pow(math.sin(k), 2.0) / (math.cos(k) / t)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64((Float64(k / l) ^ 2.0) * Float64((sin(k) ^ 2.0) / Float64(cos(k) / t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((k / l) ^ 2.0) * ((sin(k) ^ 2.0) / (cos(k) / t)));
end

Wolfram

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

Derivation

1. Initial program 32.2%

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

2. Taylor expanded in t around 0 70.7%

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

   1. times-frac 70.6%

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

4. Simplified 70.6%

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

   1. add-sqr-sqrt 70.7%

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

   2. pow2 70.7%

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

   3. div-inv 70.6%

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

   4. sqrt-prod 70.6%

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

   5. unpow2 70.6%

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

   6. sqrt-prod 41.7%

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

   7. add-sqr-sqrt 75.6%

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

   8. pow-flip 75.6%

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

   9. metadata-eval 75.6%

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

6. Applied egg-rr 75.6%

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

7. Taylor expanded in l around 0 89.8%

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

8. Taylor expanded in k around inf 70.7%

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

   1. times-frac 70.6%

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

   2. unpow2 70.6%

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

   3. unpow2 70.6%

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

   4. times-frac 89.8%

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

   5. unpow2 89.8%

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

   6. *-commutative 89.8%

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

   7. associate-/l* 89.8%

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

10. Simplified 89.8%

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

11. Final simplification 89.8%

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

Alternative 2: 80.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(k \cdot \frac{1}{\ell}\right)}^{2}\\ \mathbf{if}\;k \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{t \cdot {k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (* k (/ 1.0 l)) 2.0)))
   (if (<= k 2.2e-5)
     (/ 2.0 (* t_1 (/ (* t (pow k 2.0)) (cos k))))
     (/ 2.0 (* t_1 (/ (* t (- 0.5 (/ (cos (* 2.0 k)) 2.0))) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k * (1.0 / l)), 2.0);
	double tmp;
	if (k <= 2.2e-5) {
		tmp = 2.0 / (t_1 * ((t * pow(k, 2.0)) / cos(k)));
	} else {
		tmp = 2.0 / (t_1 * ((t * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k * (1.0d0 / l)) ** 2.0d0
    if (k <= 2.2d-5) then
        tmp = 2.0d0 / (t_1 * ((t * (k ** 2.0d0)) / cos(k)))
    else
        tmp = 2.0d0 / (t_1 * ((t * (0.5d0 - (cos((2.0d0 * k)) / 2.0d0))) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k * (1.0 / l)), 2.0);
	double tmp;
	if (k <= 2.2e-5) {
		tmp = 2.0 / (t_1 * ((t * Math.pow(k, 2.0)) / Math.cos(k)));
	} else {
		tmp = 2.0 / (t_1 * ((t * (0.5 - (Math.cos((2.0 * k)) / 2.0))) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k * (1.0 / l)), 2.0)
	tmp = 0
	if k <= 2.2e-5:
		tmp = 2.0 / (t_1 * ((t * math.pow(k, 2.0)) / math.cos(k)))
	else:
		tmp = 2.0 / (t_1 * ((t * (0.5 - (math.cos((2.0 * k)) / 2.0))) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k * Float64(1.0 / l)) ^ 2.0
	tmp = 0.0
	if (k <= 2.2e-5)
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(t * (k ^ 2.0)) / cos(k))));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(Float64(t * Float64(0.5 - Float64(cos(Float64(2.0 * k)) / 2.0))) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k * (1.0 / l)) ^ 2.0;
	tmp = 0.0;
	if (k <= 2.2e-5)
		tmp = 2.0 / (t_1 * ((t * (k ^ 2.0)) / cos(k)));
	else
		tmp = 2.0 / (t_1 * ((t * (0.5 - (cos((2.0 * k)) / 2.0))) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k * N[(1.0 / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, 2.2e-5], N[(2.0 / N[(t$95$1 * N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(N[(t * N[(0.5 - N[(N[Cos[N[(2.0 * k), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.1999999999999999e-5

   1. Initial program 34.9%

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

   2. Taylor expanded in t around 0 72.5%

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

      1. times-frac 72.9%

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

   4. Simplified 72.9%

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

      1. add-sqr-sqrt 72.9%

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

      2. pow2 72.9%

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

      3. div-inv 72.9%

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

      4. sqrt-prod 72.9%

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

      5. unpow2 72.9%

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

      6. sqrt-prod 26.2%

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

      7. add-sqr-sqrt 74.1%

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

      8. pow-flip 74.2%

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

      9. metadata-eval 74.2%

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

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in l around 0 90.5%

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

   8. Taylor expanded in k around 0 76.9%

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

   if 2.1999999999999999e-5 < k

   1. Initial program 25.6%

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

   2. Taylor expanded in t around 0 66.3%

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

      1. times-frac 65.1%

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

   4. Simplified 65.1%

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

      1. add-sqr-sqrt 65.2%

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

      2. pow2 65.2%

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

      3. div-inv 65.2%

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

      4. sqrt-prod 65.2%

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

      5. unpow2 65.2%

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

      6. sqrt-prod 79.0%

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

      7. add-sqr-sqrt 79.2%

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

      8. pow-flip 79.2%

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

      9. metadata-eval 79.2%

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

   6. Applied egg-rr 79.2%

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

   7. Taylor expanded in l around 0 88.1%

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

      1. unpow2 88.1%

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

      2. sin-mult 88.1%

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

   9. Applied egg-rr 88.1%

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

       1. div-sub 88.1%

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

       2. +-inverses 88.1%

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

       3. cos-0 88.1%

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

       4. metadata-eval 88.1%

          \[\leadsto \frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{t \cdot \left(\color{blue}{0.5} - \frac{\cos \left(k + k\right)}{2}\right)}{\cos k}} \]

       5. count-2 88.1%

          \[\leadsto \frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(2 \cdot k\right)}}{2}\right)}{\cos k}} \]

       6. *-commutative 88.1%

          \[\leadsto \frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \color{blue}{\left(k \cdot 2\right)}}{2}\right)}{\cos k}} \]

   11. Simplified 88.1%

       \[\leadsto \frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{t \cdot \color{blue}{\left(0.5 - \frac{\cos \left(k \cdot 2\right)}{2}\right)}}{\cos k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{t \cdot {k}^{2}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{1}{\ell}\right)}^{2} \cdot \frac{t \cdot \left(0.5 - \frac{\cos \left(2 \cdot k\right)}{2}\right)}{\cos k}}\\ \end{array} \]

Alternative 3: 70.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (pow (* k (/ 1.0 l)) 2.0) (/ (* t (pow k 2.0)) (cos k)))))

C

double code(double t, double l, double k) {
	return 2.0 / (pow((k * (1.0 / l)), 2.0) * ((t * pow(k, 2.0)) / cos(k)));
}

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 / (((k * (1.0d0 / l)) ** 2.0d0) * ((t * (k ** 2.0d0)) / cos(k)))
end function

Java

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

Python

def code(t, l, k):
	return 2.0 / (math.pow((k * (1.0 / l)), 2.0) * ((t * math.pow(k, 2.0)) / math.cos(k)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64((Float64(k * Float64(1.0 / l)) ^ 2.0) * Float64(Float64(t * (k ^ 2.0)) / cos(k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((k * (1.0 / l)) ^ 2.0) * ((t * (k ^ 2.0)) / cos(k)));
end

Wolfram

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

Derivation

1. Initial program 32.2%

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

2. Taylor expanded in t around 0 70.7%

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

   1. times-frac 70.6%

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

4. Simplified 70.6%

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

   1. add-sqr-sqrt 70.7%

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

   2. pow2 70.7%

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

   3. div-inv 70.6%

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

   4. sqrt-prod 70.6%

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

   5. unpow2 70.6%

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

   6. sqrt-prod 41.7%

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

   7. add-sqr-sqrt 75.6%

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

   8. pow-flip 75.6%

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

   9. metadata-eval 75.6%

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

6. Applied egg-rr 75.6%

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

7. Taylor expanded in l around 0 89.8%

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

8. Taylor expanded in k around 0 70.7%

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

9. Final simplification 70.7%

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

Alternative 4: 66.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;\frac{2}{t} \cdot {\left(\ell \cdot \sqrt{{k}^{-4}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{{\ell}^{2}} \cdot \frac{{k}^{4}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1.35e+91)
   (* (/ 2.0 t) (pow (* l (sqrt (pow k -4.0))) 2.0))
   (/ 2.0 (* (/ t (pow l 2.0)) (/ (pow k 4.0) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.35e+91) {
		tmp = (2.0 / t) * pow((l * sqrt(pow(k, -4.0))), 2.0);
	} else {
		tmp = 2.0 / ((t / pow(l, 2.0)) * (pow(k, 4.0) / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 1.35d+91) then
        tmp = (2.0d0 / t) * ((l * sqrt((k ** (-4.0d0)))) ** 2.0d0)
    else
        tmp = 2.0d0 / ((t / (l ** 2.0d0)) * ((k ** 4.0d0) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1.35e+91) {
		tmp = (2.0 / t) * Math.pow((l * Math.sqrt(Math.pow(k, -4.0))), 2.0);
	} else {
		tmp = 2.0 / ((t / Math.pow(l, 2.0)) * (Math.pow(k, 4.0) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1.35e+91:
		tmp = (2.0 / t) * math.pow((l * math.sqrt(math.pow(k, -4.0))), 2.0)
	else:
		tmp = 2.0 / ((t / math.pow(l, 2.0)) * (math.pow(k, 4.0) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1.35e+91)
		tmp = Float64(Float64(2.0 / t) * (Float64(l * sqrt((k ^ -4.0))) ^ 2.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t / (l ^ 2.0)) * Float64((k ^ 4.0) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1.35e+91)
		tmp = (2.0 / t) * ((l * sqrt((k ^ -4.0))) ^ 2.0);
	else
		tmp = 2.0 / ((t / (l ^ 2.0)) * ((k ^ 4.0) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1.35e+91], N[(N[(2.0 / t), $MachinePrecision] * N[Power[N[(l * N[Sqrt[N[Power[k, -4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, 4.0], $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.35e91

   1. Initial program 33.9%

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

      1. associate-/r* 33.9%

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

      2. associate-*l/ 34.4%

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

      3. associate--l+ 34.4%

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

   3. Simplified 34.4%

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

   4. Taylor expanded in k around 0 60.1%

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

      1. associate-*r/ 60.1%

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

      2. *-commutative 60.1%

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

      3. times-frac 60.8%

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

   6. Simplified 60.8%

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

      1. add-sqr-sqrt 60.8%

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

      2. pow2 60.8%

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

      3. div-inv 60.7%

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

      4. sqrt-prod 60.7%

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

      5. pow2 60.7%

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

      6. sqrt-prod 30.4%

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

      7. add-sqr-sqrt 69.9%

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

      8. pow-flip 69.9%

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

      9. metadata-eval 69.9%

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

   8. Applied egg-rr 69.9%

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

   if 1.35e91 < l

   1. Initial program 25.1%

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

   2. Taylor expanded in t around 0 72.0%

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

      1. associate-*r* 72.1%

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

      2. times-frac 72.0%

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

   4. Simplified 72.0%

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

   5. Taylor expanded in k around 0 64.4%

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

   6. Taylor expanded in k around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. times-frac 62.1%

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

   8. Simplified 62.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.35 \cdot 10^{+91}:\\ \;\;\;\;\frac{2}{t} \cdot {\left(\ell \cdot \sqrt{{k}^{-4}}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{{\ell}^{2}} \cdot \frac{{k}^{4}}{\cos k}}\\ \end{array} \]

Alternative 5: 66.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{t} \cdot {\left(\ell \cdot \sqrt{{k}^{-4}}\right)}^{2} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 t) (pow (* l (sqrt (pow k -4.0))) 2.0)))

C

double code(double t, double l, double k) {
	return (2.0 / t) * pow((l * sqrt(pow(k, -4.0))), 2.0);
}

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) * ((l * sqrt((k ** (-4.0d0)))) ** 2.0d0)
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / t) * Math.pow((l * Math.sqrt(Math.pow(k, -4.0))), 2.0);
}

Python

def code(t, l, k):
	return (2.0 / t) * math.pow((l * math.sqrt(math.pow(k, -4.0))), 2.0)

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / t) * (Float64(l * sqrt((k ^ -4.0))) ^ 2.0))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / t) * ((l * sqrt((k ^ -4.0))) ^ 2.0);
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / t), $MachinePrecision] * N[Power[N[(l * N[Sqrt[N[Power[k, -4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

   1. associate-/r* 32.2%

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

   2. associate-*l/ 32.6%

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

   3. associate--l+ 32.6%

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

3. Simplified 32.6%

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

4. Taylor expanded in k around 0 58.5%

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

   1. associate-*r/ 58.5%

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

   2. *-commutative 58.5%

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

   3. times-frac 58.8%

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

6. Simplified 58.8%

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

   1. add-sqr-sqrt 58.8%

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

   2. pow2 58.8%

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

   3. div-inv 58.6%

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

   4. sqrt-prod 58.6%

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

   5. pow2 58.6%

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

   6. sqrt-prod 34.2%

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

   7. add-sqr-sqrt 66.2%

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

   8. pow-flip 66.2%

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

   9. metadata-eval 66.2%

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

8. Applied egg-rr 66.2%

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

9. Final simplification 66.2%

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

Alternative 6: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}} \end{array} \]

FPCore

(FPCore (t l k) :precision binary64 (* (/ 2.0 t) (/ (pow l 2.0) (pow k 4.0))))

C

double code(double t, double l, double k) {
	return (2.0 / t) * (pow(l, 2.0) / pow(k, 4.0));
}

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) * ((l ** 2.0d0) / (k ** 4.0d0))
end function

Java

public static double code(double t, double l, double k) {
	return (2.0 / t) * (Math.pow(l, 2.0) / Math.pow(k, 4.0));
}

Python

def code(t, l, k):
	return (2.0 / t) * (math.pow(l, 2.0) / math.pow(k, 4.0))

Julia

function code(t, l, k)
	return Float64(Float64(2.0 / t) * Float64((l ^ 2.0) / (k ^ 4.0)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (2.0 / t) * ((l ^ 2.0) / (k ^ 4.0));
end

Wolfram

code[t_, l_, k_] := N[(N[(2.0 / t), $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.2%

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

   1. associate-/r* 32.2%

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

   2. associate-*l/ 32.6%

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

   3. associate--l+ 32.6%

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

3. Simplified 32.6%

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

4. Taylor expanded in k around 0 58.5%

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

   1. associate-*r/ 58.5%

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

   2. *-commutative 58.5%

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

   3. times-frac 58.8%

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

6. Simplified 58.8%

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

7. Final simplification 58.8%

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

Reproduce

herbie shell --seed 2023306 
(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))))