Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.2% → 91.1%

Time: 14.9s

Alternatives: 5

Speedup: 3.8×

• 38.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.2% 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: 91.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

   1. *-commutative 32.0%

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

   2. associate-/r* 32.0%

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

3. Simplified 37.9%

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

   1. clear-num 37.9%

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

   2. +-rgt-identity 37.9%

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

   3. associate-/r/ 37.9%

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

   4. pow-flip 38.5%

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

   5. metadata-eval 38.5%

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

6. Applied egg-rr 38.5%

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

7. Taylor expanded in k around inf 69.7%

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

   1. times-frac 71.0%

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

   2. *-commutative 71.0%

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

9. Simplified 71.0%

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

    1. add-sqr-sqrt 71.0%

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

    2. pow2 71.0%

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

    3. div-inv 70.6%

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

    4. sqrt-prod 70.6%

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

    5. sqrt-pow1 81.9%

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

    6. metadata-eval 81.9%

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

    7. pow1 81.9%

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

    8. pow-flip 82.3%

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

    9. metadata-eval 82.3%

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

11. Applied egg-rr 82.3%

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

12. Taylor expanded in l around 0 71.0%

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

    1. unpow2 71.0%

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

    2. unpow2 71.0%

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

    3. times-frac 93.0%

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

    4. *-lft-identity 93.0%

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

    5. associate-*l/ 93.0%

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

    6. *-lft-identity 93.0%

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

    7. associate-*l/ 93.0%

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

    8. unpow2 93.0%

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

    9. associate-*l/ 93.0%

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

    10. *-lft-identity 93.0%

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

14. Simplified 93.0%

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

Alternative 2: 91.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \ell \cdot \frac{1}{k}\\ 2 \cdot \left(\frac{\cos k}{{\sin k}^{2} \cdot t} \cdot \left(t\_1 \cdot t\_1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* l (/ 1.0 k))))
   (* 2.0 (* (/ (cos k) (* (pow (sin k) 2.0) t)) (* t_1 t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = l * (1.0 / k);
	return 2.0 * ((cos(k) / (pow(sin(k), 2.0) * t)) * (t_1 * t_1));
}

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
    t_1 = l * (1.0d0 / k)
    code = 2.0d0 * ((cos(k) / ((sin(k) ** 2.0d0) * t)) * (t_1 * t_1))
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l * (1.0 / k);
	return 2.0 * ((Math.cos(k) / (Math.pow(Math.sin(k), 2.0) * t)) * (t_1 * t_1));
}

Python

def code(t, l, k):
	t_1 = l * (1.0 / k)
	return 2.0 * ((math.cos(k) / (math.pow(math.sin(k), 2.0) * t)) * (t_1 * t_1))

Julia

function code(t, l, k)
	t_1 = Float64(l * Float64(1.0 / k))
	return Float64(2.0 * Float64(Float64(cos(k) / Float64((sin(k) ^ 2.0) * t)) * Float64(t_1 * t_1)))
end

MATLAB

function tmp = code(t, l, k)
	t_1 = l * (1.0 / k);
	tmp = 2.0 * ((cos(k) / ((sin(k) ^ 2.0) * t)) * (t_1 * t_1));
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 32.0%

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

   1. *-commutative 32.0%

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

   2. associate-/r* 32.0%

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

3. Simplified 37.9%

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

   1. clear-num 37.9%

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

   2. +-rgt-identity 37.9%

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

   3. associate-/r/ 37.9%

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

   4. pow-flip 38.5%

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

   5. metadata-eval 38.5%

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

6. Applied egg-rr 38.5%

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

7. Taylor expanded in k around inf 69.7%

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

   1. times-frac 71.0%

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

   2. *-commutative 71.0%

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

9. Simplified 71.0%

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

    1. add-sqr-sqrt 71.0%

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

    2. pow2 71.0%

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

    3. div-inv 70.6%

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

    4. sqrt-prod 70.6%

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

    5. sqrt-pow1 81.9%

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

    6. metadata-eval 81.9%

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

    7. pow1 81.9%

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

    8. pow-flip 82.3%

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

    9. metadata-eval 82.3%

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

11. Applied egg-rr 82.3%

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

    1. unpow2 82.3%

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

    2. *-commutative 82.3%

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

    3. sqrt-pow1 54.7%

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

    4. metadata-eval 54.7%

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

    5. inv-pow 54.7%

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

    6. *-commutative 54.7%

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

    7. sqrt-pow1 93.0%

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

    8. metadata-eval 93.0%

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

    9. inv-pow 93.0%

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

13. Applied egg-rr 93.0%

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

14. Final simplification 93.0%

    \[\leadsto 2 \cdot \left(\frac{\cos k}{{\sin k}^{2} \cdot t} \cdot \left(\left(\ell \cdot \frac{1}{k}\right) \cdot \left(\ell \cdot \frac{1}{k}\right)\right)\right) \]
15. Add Preprocessing

Alternative 3: 70.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{1}{t \cdot {k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 4.2e-159)
   (* l (* 2.0 (/ (/ l (pow k 4.0)) t)))
   (* 2.0 (* (/ (pow l 2.0) (pow k 2.0)) (/ 1.0 (* t (pow k 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.2e-159) {
		tmp = l * (2.0 * ((l / pow(k, 4.0)) / t));
	} else {
		tmp = 2.0 * ((pow(l, 2.0) / pow(k, 2.0)) * (1.0 / (t * pow(k, 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 <= 4.2d-159) then
        tmp = l * (2.0d0 * ((l / (k ** 4.0d0)) / t))
    else
        tmp = 2.0d0 * (((l ** 2.0d0) / (k ** 2.0d0)) * (1.0d0 / (t * (k ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.2e-159) {
		tmp = l * (2.0 * ((l / Math.pow(k, 4.0)) / t));
	} else {
		tmp = 2.0 * ((Math.pow(l, 2.0) / Math.pow(k, 2.0)) * (1.0 / (t * Math.pow(k, 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 4.2e-159:
		tmp = l * (2.0 * ((l / math.pow(k, 4.0)) / t))
	else:
		tmp = 2.0 * ((math.pow(l, 2.0) / math.pow(k, 2.0)) * (1.0 / (t * math.pow(k, 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 4.2e-159)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) / t)));
	else
		tmp = Float64(2.0 * Float64(Float64((l ^ 2.0) / (k ^ 2.0)) * Float64(1.0 / Float64(t * (k ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 4.2e-159)
		tmp = l * (2.0 * ((l / (k ^ 4.0)) / t));
	else
		tmp = 2.0 * (((l ^ 2.0) / (k ^ 2.0)) * (1.0 / (t * (k ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 4.2e-159], N[(l * N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.1999999999999998e-159

   1. Initial program 32.4%

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

   3. Simplified 38.7%

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

   5. Taylor expanded in k around 0 61.2%

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

      1. add-exp-log 49.1%

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

      2. *-commutative 49.1%

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

      3. pow2 49.1%

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

   7. Applied egg-rr 49.1%

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

      1. associate-/r* 49.1%

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

      2. pow2 49.1%

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

      3. associate-*r* 56.0%

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

      4. add-exp-log 72.7%

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

      5. div-inv 72.7%

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

      6. pow-flip 72.7%

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

      7. metadata-eval 72.7%

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

   9. Applied egg-rr 72.7%

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

   10. Taylor expanded in t around 0 72.7%

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

       1. associate-/r* 73.5%

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

   12. Simplified 73.5%

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

   if 4.1999999999999998e-159 < l

   1. Initial program 31.1%

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

      1. *-commutative 31.1%

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

      2. associate-/r* 31.1%

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

   3. Simplified 39.4%

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

      1. clear-num 39.4%

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

      2. +-rgt-identity 39.4%

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

      3. associate-/r/ 39.4%

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

      4. pow-flip 41.3%

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

      5. metadata-eval 41.3%

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

   6. Applied egg-rr 41.3%

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

   7. Taylor expanded in k around inf 76.0%

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

      1. times-frac 77.1%

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

      2. *-commutative 77.1%

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

   9. Simplified 77.1%

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

   10. Taylor expanded in k around 0 59.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.2 \cdot 10^{-159}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\ell}{{k}^{4}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{1}{t \cdot {k}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

3. Simplified 39.3%

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

5. Taylor expanded in k around 0 58.6%

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

   1. add-exp-log 43.1%

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

   2. *-commutative 43.1%

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

   3. pow2 43.1%

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

7. Applied egg-rr 43.1%

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

   1. associate-/r* 43.1%

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

   2. pow2 43.1%

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

   3. associate-*r* 48.3%

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

   4. add-exp-log 66.9%

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

   5. div-inv 66.8%

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

   6. pow-flip 66.8%

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

   7. metadata-eval 66.8%

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

9. Applied egg-rr 66.8%

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

10. Taylor expanded in t around 0 66.9%

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

    1. associate-/r* 67.1%

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

12. Simplified 67.1%

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

13. Final simplification 67.1%

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

Alternative 5: 68.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.0%

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

3. Simplified 39.3%

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

5. Taylor expanded in k around 0 58.6%

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

   1. add-exp-log 43.1%

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

   2. *-commutative 43.1%

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

   3. pow2 43.1%

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

7. Applied egg-rr 43.1%

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

   1. associate-/r* 43.1%

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

   2. pow2 43.1%

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

   3. associate-*r* 48.3%

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

   4. add-exp-log 66.9%

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

   5. div-inv 66.8%

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

   6. pow-flip 66.8%

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

   7. metadata-eval 66.8%

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

9. Applied egg-rr 66.8%

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

10. Taylor expanded in t around 0 66.9%

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

11. Final simplification 66.9%

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

Reproduce

herbie shell --seed 2024094 
(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))))