Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.5% → 92.7%

Time: 20.5s

Alternatives: 9

Speedup: 3.8×

• 35.4% 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: 34.5% 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: 92.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-/r* 36.3%

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

   2. associate-*l/ 37.0%

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

   3. associate--l+ 37.0%

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

3. Simplified 37.0%

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

5. Applied egg-rr 21.9%

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

   1. expm1-def 24.4%

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

   2. expm1-log1p 24.6%

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

   3. associate-*l* 24.7%

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

   4. *-commutative 24.7%

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

7. Simplified 24.7%

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

8. Taylor expanded in k around inf 43.9%

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

   1. associate-/l* 44.7%

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

   2. associate-/r/ 44.7%

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

10. Simplified 44.7%

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

    1. unpow-prod-down 42.6%

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

    2. *-commutative 42.6%

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

    3. pow2 42.6%

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

    4. add-sqr-sqrt 93.8%

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

12. Applied egg-rr 93.8%

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

    1. associate-*r/ 93.9%

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

14. Applied egg-rr 93.9%

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

15. Final simplification 93.9%

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

Alternative 2: 92.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-/r* 36.3%

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

   2. associate-*l/ 37.0%

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

   3. associate--l+ 37.0%

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

3. Simplified 37.0%

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

5. Applied egg-rr 21.9%

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

   1. expm1-def 24.4%

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

   2. expm1-log1p 24.6%

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

   3. associate-*l* 24.7%

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

   4. *-commutative 24.7%

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

7. Simplified 24.7%

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

8. Taylor expanded in k around inf 43.9%

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

   1. associate-/l* 44.7%

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

   2. associate-/r/ 44.7%

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

10. Simplified 44.7%

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

    1. expm1-log1p-u 43.9%

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

    2. expm1-udef 35.9%

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

    3. *-commutative 35.9%

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

    4. unpow-prod-down 34.6%

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

    5. pow2 34.6%

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

    6. add-sqr-sqrt 35.2%

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

    7. *-commutative 35.2%

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

12. Applied egg-rr 35.2%

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

    1. expm1-def 74.8%

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

    2. expm1-log1p 93.8%

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

    3. associate-*r/ 93.4%

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

    4. *-commutative 93.4%

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

    5. associate-/l* 93.8%

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

14. Simplified 93.8%

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

15. Final simplification 93.8%

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

Alternative 3: 92.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-/r* 36.3%

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

   2. associate-*l/ 37.0%

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

   3. associate--l+ 37.0%

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

3. Simplified 37.0%

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

5. Applied egg-rr 21.9%

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

   1. expm1-def 24.4%

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

   2. expm1-log1p 24.6%

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

   3. associate-*l* 24.7%

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

   4. *-commutative 24.7%

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

7. Simplified 24.7%

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

8. Taylor expanded in k around inf 43.9%

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

   1. associate-/l* 44.7%

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

   2. associate-/r/ 44.7%

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

10. Simplified 44.7%

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

    1. unpow-prod-down 42.6%

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

    2. *-commutative 42.6%

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

    3. pow2 42.6%

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

    4. add-sqr-sqrt 93.8%

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

12. Applied egg-rr 93.8%

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

13. Final simplification 93.8%

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

Alternative 4: 73.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

   1. associate-/r* 36.3%

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

   2. associate-*l/ 37.0%

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

   3. associate--l+ 37.0%

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

3. Simplified 37.0%

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

5. Applied egg-rr 21.9%

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

   1. expm1-def 24.4%

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

   2. expm1-log1p 24.6%

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

   3. associate-*l* 24.7%

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

   4. *-commutative 24.7%

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

7. Simplified 24.7%

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

8. Taylor expanded in k around inf 43.9%

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

   1. associate-/l* 44.7%

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

   2. associate-/r/ 44.7%

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

10. Simplified 44.7%

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

    1. unpow-prod-down 42.6%

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

    2. *-commutative 42.6%

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

    3. pow2 42.6%

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

    4. add-sqr-sqrt 93.8%

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

12. Applied egg-rr 93.8%

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

13. Taylor expanded in k around 0 75.7%

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

14. Final simplification 75.7%

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

Alternative 5: 60.5% 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 36.8%

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

   1. associate-/r* 36.3%

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

   2. associate-*l/ 37.0%

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

   3. associate--l+ 37.0%

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

3. Simplified 37.0%

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

4. Taylor expanded in k around 0 62.1%

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

5. Final simplification 62.1%

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

Alternative 6: 60.5% accurate, 2.0× speedup

Math

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

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 36.8%

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

3. Simplified 42.5%

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

   1. unpow3 42.5%

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

   2. times-frac 52.2%

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

   3. pow2 52.2%

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

5. Applied egg-rr 52.2%

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

6. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/l* 61.8%

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

8. Simplified 61.8%

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

   1. associate-/r/ 61.8%

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

   2. div-inv 61.8%

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

   3. pow-flip 62.2%

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

   4. metadata-eval 62.2%

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

10. Applied egg-rr 62.2%

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

11. Final simplification 62.2%

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

Alternative 7: 60.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Simplified 42.5%

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

   1. unpow3 42.5%

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

   2. times-frac 52.2%

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

   3. pow2 52.2%

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

5. Applied egg-rr 52.2%

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

6. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/l* 61.8%

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

8. Simplified 61.8%

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

   1. expm1-log1p-u 38.4%

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

   2. expm1-udef 36.8%

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

   3. associate-/r/ 36.8%

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

   4. div-inv 36.8%

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

   5. pow-flip 37.2%

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

   6. metadata-eval 37.2%

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

10. Applied egg-rr 37.2%

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

    1. expm1-def 38.8%

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

    2. expm1-log1p 62.2%

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

    3. *-commutative 62.2%

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

    4. associate-*l* 62.5%

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

12. Simplified 62.5%

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

13. Final simplification 62.5%

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

Alternative 8: 70.2% 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 36.8%

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

3. Simplified 42.5%

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

   1. unpow3 42.5%

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

   2. times-frac 52.2%

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

   3. pow2 52.2%

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

5. Applied egg-rr 52.2%

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

6. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/l* 61.8%

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

8. Simplified 61.8%

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

   1. add-sqr-sqrt 61.8%

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

   2. pow2 61.8%

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

   3. div-inv 61.8%

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

   4. sqrt-prod 61.8%

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

   5. pow2 61.8%

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

   6. sqrt-prod 34.5%

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

   7. add-sqr-sqrt 69.3%

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

   8. pow-flip 69.3%

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

   9. metadata-eval 69.3%

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

   10. metadata-eval 69.3%

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

   11. pow-prod-up 69.3%

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

   12. sqrt-unprod 72.3%

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

   13. add-sqr-sqrt 72.3%

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

10. Applied egg-rr 72.3%

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

11. Final simplification 72.3%

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

Alternative 9: 60.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.8%

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

3. Simplified 42.5%

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

   1. unpow3 42.5%

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

   2. times-frac 52.2%

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

   3. pow2 52.2%

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

5. Applied egg-rr 52.2%

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

6. Taylor expanded in k around 0 62.1%

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

   1. *-commutative 62.1%

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

   2. associate-/l* 61.8%

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

8. Simplified 61.8%

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

   1. pow2 61.8%

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

10. Applied egg-rr 61.8%

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

11. Final simplification 61.8%

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

Reproduce

herbie shell --seed 2023318 
(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))))