Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.8% → 93.0%

Time: 21.6s

Alternatives: 9

Speedup: 2.0×

• 34.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 34.8% 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: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{2 \cdot \frac{\cos k}{{\left(\frac{k}{\ell}\right)}^{2}}}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t -1e-311)
   (/ (* 2.0 (/ (cos k) (pow (/ k l) 2.0))) (* t (pow (sin k) 2.0)))
   (/ 2.0 (* (* (sin k) (tan k)) (pow (* (/ k l) (sqrt t)) 2.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= -1e-311) {
		tmp = (2.0 * (cos(k) / pow((k / l), 2.0))) / (t * pow(sin(k), 2.0));
	} else {
		tmp = 2.0 / ((sin(k) * tan(k)) * pow(((k / l) * sqrt(t)), 2.0));
	}
	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 (t <= (-1d-311)) then
        tmp = (2.0d0 * (cos(k) / ((k / l) ** 2.0d0))) / (t * (sin(k) ** 2.0d0))
    else
        tmp = 2.0d0 / ((sin(k) * tan(k)) * (((k / l) * sqrt(t)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1e-311) {
		tmp = (2.0 * (Math.cos(k) / Math.pow((k / l), 2.0))) / (t * Math.pow(Math.sin(k), 2.0));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * Math.pow(((k / l) * Math.sqrt(t)), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= -1e-311:
		tmp = (2.0 * (math.cos(k) / math.pow((k / l), 2.0))) / (t * math.pow(math.sin(k), 2.0))
	else:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * math.pow(((k / l) * math.sqrt(t)), 2.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= -1e-311)
		tmp = Float64(Float64(2.0 * Float64(cos(k) / (Float64(k / l) ^ 2.0))) / Float64(t * (sin(k) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * (Float64(Float64(k / l) * sqrt(t)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1e-311)
		tmp = (2.0 * (cos(k) / ((k / l) ^ 2.0))) / (t * (sin(k) ^ 2.0));
	else
		tmp = 2.0 / ((sin(k) * tan(k)) * (((k / l) * sqrt(t)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, -1e-311], N[(N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(k / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.99999999999948e-312

   1. Initial program 31.5%

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

   3. Simplified 40.3%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 0.0%

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

      2. expm1-log1p 0.0%

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

      3. associate-*l* 0.0%

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

      4. *-commutative 0.0%

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

   7. Simplified 0.0%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-udef 0.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\right)} - 1} \]

      3. pow-prod-down 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}}\right)} - 1 \]

      4. pow2 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

   9. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 0.0%

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

       2. expm1-log1p 0.0%

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

       3. times-frac 0.0%

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

       4. associate-/l* 0.0%

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

       5. associate-/r/ 0.0%

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

   11. Simplified 0.0%

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

   12. Taylor expanded in k around 0 0.0%

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

       1. associate-/l/ 0.0%

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

   14. Simplified 0.0%

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

   15. Taylor expanded in k around inf 72.2%

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

       1. associate-/r* 72.4%

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

       2. associate-*r/ 72.4%

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

       3. *-commutative 72.4%

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

       4. associate-/l* 72.4%

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

       5. unpow2 72.4%

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

       6. unpow2 72.4%

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

       7. times-frac 89.5%

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

       8. unpow2 89.5%

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

   17. Simplified 89.5%

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

   if -9.99999999999948e-312 < t

   1. Initial program 30.2%

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

   3. Simplified 38.9%

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

   5. Applied egg-rr 37.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 46.1%

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

      2. expm1-log1p 46.4%

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

      3. associate-*l* 46.4%

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

      4. *-commutative 46.4%

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

   7. Simplified 46.4%

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

      1. expm1-log1p-u 46.1%

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

      2. expm1-udef 37.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\right)} - 1} \]

      3. pow-prod-down 37.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}}\right)} - 1 \]

      4. pow2 37.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

      5. add-sqr-sqrt 56.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

   9. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 67.5%

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

       2. expm1-log1p 69.5%

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

       3. times-frac 61.9%

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

       4. associate-/l* 64.3%

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

       5. associate-/r/ 62.2%

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

   11. Simplified 62.2%

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

   12. Taylor expanded in k around 0 96.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-311}:\\ \;\;\;\;\frac{2 \cdot \frac{\cos k}{{\left(\frac{k}{\ell}\right)}^{2}}}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 2: 92.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin k \cdot \tan k\\ \mathbf{if}\;t \leq 1.2 \cdot 10^{-305}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot {\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (sin k) (tan k))))
   (if (<= t 1.2e-305)
     (/ 2.0 (* (pow (/ k l) 2.0) (* t t_1)))
     (/ 2.0 (* t_1 (pow (* (/ k l) (sqrt t)) 2.0))))))

C

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double tmp;
	if (t <= 1.2e-305) {
		tmp = 2.0 / (pow((k / l), 2.0) * (t * t_1));
	} else {
		tmp = 2.0 / (t_1 * pow(((k / l) * sqrt(t)), 2.0));
	}
	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 = sin(k) * tan(k)
    if (t <= 1.2d-305) then
        tmp = 2.0d0 / (((k / l) ** 2.0d0) * (t * t_1))
    else
        tmp = 2.0d0 / (t_1 * (((k / l) * sqrt(t)) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double tmp;
	if (t <= 1.2e-305) {
		tmp = 2.0 / (Math.pow((k / l), 2.0) * (t * t_1));
	} else {
		tmp = 2.0 / (t_1 * Math.pow(((k / l) * Math.sqrt(t)), 2.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.sin(k) * math.tan(k)
	tmp = 0
	if t <= 1.2e-305:
		tmp = 2.0 / (math.pow((k / l), 2.0) * (t * t_1))
	else:
		tmp = 2.0 / (t_1 * math.pow(((k / l) * math.sqrt(t)), 2.0))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(sin(k) * tan(k))
	tmp = 0.0
	if (t <= 1.2e-305)
		tmp = Float64(2.0 / Float64((Float64(k / l) ^ 2.0) * Float64(t * t_1)));
	else
		tmp = Float64(2.0 / Float64(t_1 * (Float64(Float64(k / l) * sqrt(t)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) * tan(k);
	tmp = 0.0;
	if (t <= 1.2e-305)
		tmp = 2.0 / (((k / l) ^ 2.0) * (t * t_1));
	else
		tmp = 2.0 / (t_1 * (((k / l) * sqrt(t)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 1.2e-305], N[(2.0 / N[(N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision] * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[Power[N[(N[(k / l), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.2000000000000001e-305

   1. Initial program 31.3%

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

   3. Simplified 40.0%

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

   5. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 0.0%

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

      2. expm1-log1p 0.0%

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

      3. associate-*l* 0.0%

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

      4. *-commutative 0.0%

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

   7. Simplified 0.0%

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

   8. Taylor expanded in k around 0 0.7%

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

      1. expm1-log1p-u 0.6%

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

      2. expm1-udef 0.6%

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

      3. unpow-prod-down 0.6%

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

      4. pow2 0.6%

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

      5. add-sqr-sqrt 0.6%

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

      6. *-commutative 0.6%

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

      7. unpow-prod-down 0.6%

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

      8. pow2 0.6%

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

      9. add-sqr-sqrt 29.5%

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

   10. Applied egg-rr 29.5%

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

       1. expm1-def 38.3%

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

       2. expm1-log1p 87.5%

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

       3. associate-*r* 89.5%

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

   12. Simplified 89.5%

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

   if 1.2000000000000001e-305 < t

   1. Initial program 30.4%

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

   3. Simplified 39.2%

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

   5. Applied egg-rr 37.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 46.5%

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

      2. expm1-log1p 46.8%

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

      3. associate-*l* 46.8%

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

      4. *-commutative 46.8%

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

   7. Simplified 46.8%

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

      1. expm1-log1p-u 46.5%

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

      2. expm1-udef 37.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\sqrt{\sin k \cdot \tan k} \cdot \left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\right)} - 1} \]

      3. pow-prod-down 37.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{{\left(\sqrt{\sin k \cdot \tan k}\right)}^{2} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}}\right)} - 1 \]

      4. pow2 37.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sqrt{\sin k \cdot \tan k} \cdot \sqrt{\sin k \cdot \tan k}\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

      5. add-sqr-sqrt 56.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right)} \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1 \]

   9. Applied egg-rr 56.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t} \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 68.1%

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

       2. expm1-log1p 70.1%

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

       3. times-frac 62.4%

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

       4. associate-/l* 64.8%

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

       5. associate-/r/ 62.7%

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

   11. Simplified 62.7%

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

   12. Taylor expanded in k around 0 95.9%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-305}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{\ell}\right)}^{2} \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{\ell} \cdot \sqrt{t}\right)}^{2}}\\ \end{array} \]

Alternative 3: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-139}:\\ \;\;\;\;\frac{2 \cdot {\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 1e-139)
   (/ (* 2.0 (pow (* l (pow k -2.0)) 2.0)) t)
   (/ 2.0 (* (* (sin k) (tan k)) (* t (pow (/ k l) 2.0))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 1e-139) {
		tmp = (2.0 * pow((l * pow(k, -2.0)), 2.0)) / t;
	} else {
		tmp = 2.0 / ((sin(k) * tan(k)) * (t * pow((k / l), 2.0)));
	}
	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 <= 1d-139) then
        tmp = (2.0d0 * ((l * (k ** (-2.0d0))) ** 2.0d0)) / t
    else
        tmp = 2.0d0 / ((sin(k) * tan(k)) * (t * ((k / l) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 1e-139) {
		tmp = (2.0 * Math.pow((l * Math.pow(k, -2.0)), 2.0)) / t;
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * (t * Math.pow((k / l), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 1e-139:
		tmp = (2.0 * math.pow((l * math.pow(k, -2.0)), 2.0)) / t
	else:
		tmp = 2.0 / ((math.sin(k) * math.tan(k)) * (t * math.pow((k / l), 2.0)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 1e-139)
		tmp = Float64(Float64(2.0 * (Float64(l * (k ^ -2.0)) ^ 2.0)) / t);
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(t * (Float64(k / l) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 1e-139)
		tmp = (2.0 * ((l * (k ^ -2.0)) ^ 2.0)) / t;
	else
		tmp = 2.0 / ((sin(k) * tan(k)) * (t * ((k / l) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 1e-139], N[(N[(2.0 * N[Power[N[(l * N[Power[k, -2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(t * N[Power[N[(k / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.00000000000000003e-139

   1. Initial program 31.8%

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

   3. Simplified 41.4%

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

   4. Taylor expanded in k around 0 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-/r* 59.3%

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

   6. Simplified 59.3%

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

      1. div-inv 59.3%

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

      2. pow-flip 59.3%

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

      3. metadata-eval 59.3%

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

   8. Applied egg-rr 59.3%

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

      1. associate-*l/ 60.9%

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

   10. Applied egg-rr 60.9%

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

       1. associate-*r/ 60.9%

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

       2. add-sqr-sqrt 60.9%

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

       3. pow2 60.9%

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

       4. sqrt-prod 60.9%

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

       5. unpow2 60.9%

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

       6. sqrt-prod 16.9%

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

       7. add-sqr-sqrt 69.9%

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

       8. sqrt-pow1 73.4%

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

       9. metadata-eval 73.4%

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

   12. Applied egg-rr 73.4%

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

   if 1.00000000000000003e-139 < l

   1. Initial program 29.1%

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

   3. Simplified 36.1%

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

   5. Applied egg-rr 16.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 20.7%

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

      2. expm1-log1p 20.9%

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

      3. associate-*l* 20.9%

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

      4. *-commutative 20.9%

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

   7. Simplified 20.9%

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

   8. Taylor expanded in k around 0 34.3%

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

      1. expm1-log1p-u 33.9%

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

      2. expm1-udef 25.4%

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

      3. unpow-prod-down 25.4%

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

      4. pow2 25.4%

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

      5. add-sqr-sqrt 30.2%

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

      6. *-commutative 30.2%

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

      7. unpow-prod-down 30.2%

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

      8. pow2 30.2%

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

      9. add-sqr-sqrt 40.8%

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

   10. Applied egg-rr 40.8%

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

       1. expm1-def 57.0%

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

       2. expm1-log1p 93.2%

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

   12. Simplified 93.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{-139}:\\ \;\;\;\;\frac{2 \cdot {\left(\ell \cdot {k}^{-2}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot {\left(\frac{k}{\ell}\right)}^{2}\right)}\\ \end{array} \]

Alternative 4: 90.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 / (pow((k / l), 2.0) * (t * (sin(k) * tan(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 / l) ** 2.0d0) * (t * (sin(k) * tan(k))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Simplified 39.6%

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

5. Applied egg-rr 17.6%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{{\left(\left(\sqrt{\tan k \cdot \sin k} \cdot \frac{k}{t}\right) \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2}}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 21.6%

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

   2. expm1-log1p 21.7%

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

   3. associate-*l* 21.7%

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

   4. *-commutative 21.7%

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

7. Simplified 21.7%

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

8. Taylor expanded in k around 0 28.7%

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

   1. expm1-log1p-u 28.4%

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

   2. expm1-udef 23.0%

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

   3. unpow-prod-down 23.0%

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

   4. pow2 23.0%

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

   5. add-sqr-sqrt 32.3%

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

   6. *-commutative 32.3%

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

   7. unpow-prod-down 31.7%

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

   8. pow2 31.7%

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

   9. add-sqr-sqrt 47.1%

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

10. Applied egg-rr 47.1%

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

    1. expm1-def 58.3%

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

    2. expm1-log1p 88.6%

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

    3. associate-*r* 90.0%

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

12. Simplified 90.0%

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

13. Final simplification 90.0%

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

Alternative 5: 59.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 * (pow(k, -4.0) * (pow(l, 2.0) / 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 ** (-4.0d0)) * ((l ** 2.0d0) / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Simplified 39.6%

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

4. Taylor expanded in k around 0 60.2%

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

   1. *-commutative 60.2%

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

   2. associate-/r* 58.2%

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

6. Simplified 58.2%

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

   1. div-inv 58.2%

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

   2. pow-flip 58.2%

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

   3. metadata-eval 58.2%

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

8. Applied egg-rr 58.2%

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

9. Final simplification 58.2%

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

Alternative 6: 60.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 * ((pow(l, 2.0) * pow(k, -4.0)) / 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 * (((l ** 2.0d0) * (k ** (-4.0d0))) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Simplified 39.6%

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

4. Taylor expanded in k around 0 60.2%

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

   1. *-commutative 60.2%

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

   2. associate-/r* 58.2%

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

6. Simplified 58.2%

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

   1. div-inv 58.2%

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

   2. pow-flip 58.2%

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

   3. metadata-eval 58.2%

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

8. Applied egg-rr 58.2%

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

   1. associate-*l/ 59.8%

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

10. Applied egg-rr 59.8%

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

11. Final simplification 59.8%

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

Alternative 7: 60.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 * (pow(l, 2.0) / (t * 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 * ((l ** 2.0d0) / (t * (k ** 4.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Simplified 39.6%

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

4. Taylor expanded in k around 0 60.2%

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

5. Final simplification 60.2%

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

Alternative 8: 69.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return 2.0 / (t / pow((l * pow(k, -2.0)), 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 * (k ** (-2.0d0))) ** 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Simplified 39.6%

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

4. Taylor expanded in k around 0 60.2%

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

   1. *-commutative 60.2%

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

   2. associate-/r* 58.2%

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

6. Simplified 58.2%

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

   1. div-inv 58.2%

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

   2. pow-flip 58.2%

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

   3. metadata-eval 58.2%

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

8. Applied egg-rr 58.2%

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

   1. associate-*l/ 59.8%

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

10. Applied egg-rr 59.8%

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

    1. expm1-log1p-u 41.6%

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

    2. expm1-udef 40.3%

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

    3. associate-*r/ 40.3%

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

    4. add-sqr-sqrt 40.3%

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

    5. pow2 40.3%

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

    6. sqrt-prod 40.3%

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

    7. unpow2 40.3%

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

    8. sqrt-prod 19.7%

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

    9. add-sqr-sqrt 43.3%

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

    10. sqrt-pow1 43.4%

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

    11. metadata-eval 43.4%

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

12. Applied egg-rr 43.4%

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

    1. expm1-def 46.4%

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

    2. expm1-log1p 68.2%

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

    3. associate-/l* 68.2%

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

14. Simplified 68.2%

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

15. Final simplification 68.2%

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

Alternative 9: 69.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return (2.0 * pow((l * pow(k, -2.0)), 2.0)) / 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 * ((l * (k ** (-2.0d0))) ** 2.0d0)) / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.9%

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

3. Simplified 39.6%

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

4. Taylor expanded in k around 0 60.2%

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

   1. *-commutative 60.2%

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

   2. associate-/r* 58.2%

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

6. Simplified 58.2%

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

   1. div-inv 58.2%

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

   2. pow-flip 58.2%

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

   3. metadata-eval 58.2%

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

8. Applied egg-rr 58.2%

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

   1. associate-*l/ 59.8%

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

10. Applied egg-rr 59.8%

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

    1. associate-*r/ 59.8%

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

    2. add-sqr-sqrt 59.8%

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

    3. pow2 59.8%

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

    4. sqrt-prod 59.8%

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

    5. unpow2 59.8%

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

    6. sqrt-prod 30.9%

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

    7. add-sqr-sqrt 65.9%

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

    8. sqrt-pow1 68.2%

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

    9. metadata-eval 68.2%

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

12. Applied egg-rr 68.2%

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

13. Final simplification 68.2%

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

Reproduce

herbie shell --seed 2023300 
(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))))